VTRIVSQVE ARITHMET fCE snbsp;Epitome ƒ X narijs autho.^nbsp;ribus concinnatapernbsp;HVdalrichumnbsp;Rcgium.

lt;?/«£)

m.d.xxxvi.

GALLVS M A RIV 8 candido Ledori S ? D.

VEMadpnodujm apud jnaiorcs noftros,candide Ledor, dodos fem-per flonu'fle legimusjnbsp;qiii, ne fuae erudiet oniinbsp;monumenta quædajnnbsp;»nterircnt,non ob leucmpopulan's aurxnbsp;glon'â captandâjfed in utilitatem eorurnnbsp;quorum animus Cadto naturae impetu adnbsp;liberalia ftudia rapiebatur,præclarafuinbsp;«ngenîj opera ÔCfuis amp; poften's relique-runt : ica ÔC hodie ui'ros in omnibus difei-plinis excellentiflimos uidemus,qui tar-dis Sc infelicibus quorundam ingenijsnbsp;fiibuenire cupientes,immodicos fuac do-lt;^rinaclabores literis mandate uoluerScnbsp;Qua in re fingularem animi manfuetudinbsp;nein, uoluncariamcp proinouendæ luuennbsp;tucis operam cernere licet, c5 quod Her-A if

Præfatio.

cuîCIS laboribus eft partum, tanta ïibera» Il rare ad omnes cmanat« Nam à naturanbsp;i4 ui'ti'j pluribus cft iniïtum, ut quod in-genij indufîria.funt afTecutijid curiofe nenbsp;dicam auare penitus rccondunt, quogionbsp;hôfîus pcculiarem oftentando artem launbsp;dem ueiîart poflînt. Hoc itaq; maiori di-ligcnttagratum oftendere decct am'mû,nbsp;quo promptius ab aliquo bcneficiû pro-ficifcitur.Eam ob rcmmaion'bus noftrisnbsp;omnium difeiplinarum peritifl’imisplu-rimum debemus,quibus prima cura fuitnbsp;ut nos fuis uigiltjs lucubrationibuscp iu-uarcnt,quodetiam alacriuultu, ferenanbsp;fronteae fpontaneo fane animo feccrût.nbsp;In quorum albo Huldrichum RcgiuiTlnbsp;artium liberaiium indagatorem folertifnbsp;fimum,fub quo preceptore cuin ego tùmnbsp;plurcsaHj in fjs ipfis plurimum profeci'nbsp;mus non immerito numerandumarbi-tror. Namutcætcrasin co artes præte-ream,pra:fenshocde Arithmetica opusnbsp;in

Præfatio.

in hicrm emittcre iioluic, quo airqug fâl-tem utilicas ad omnes huius ara'sfiu l*o-fos rcdiret, quanquam complurcs Ctiam alïos de hac difdplina accuratiffîmc con-fcnpfiffe animaduertcnt * Verum cumnbsp;alrjardiia quædam fubln-nia,imo ab-din'ffîma numerï myftcria excufïênnt,nbsp;quidam uero brcuitatem fedtanies ,mùxnbsp;fpaciofbs huius artis limites contraxe-nnt,adeoutpigriora obtufiorab? ingénia nullum fere indefperarepoITmt fm-dum, rede profecto egiffe putamus, qgt;nbsp;hoc inuento dccreuerit infirmo quorun-damintelledui efle confulen dum. Namnbsp;multis natura ingcnîj præbuit fagacita-tcm intelleduscg acrimoniam, utea in-terdum aflequahtur quæ humante mentis capacitatcm longe cxccdunt antccel-luncq?, quibus non opus eft laboriofa mnbsp;hifcc rebus uti in dagine. Comp 1 ur es ue-to rcperias qupseadem natura neglecftuinbsp;habuiflc uideacur, fiin addifcendis arti-

A iîj

Præfatl'o.

bus fpeAsjUcris acumen am'rni: ab his nihil utilicatis nifipracmanfum in os infc-ratur,eftfperandum* Qiiosmeritogra-tulari iubeo, quod in ipforum gratiam, hoc de numeri fcientia opiK tanto ftudionbsp;fit congeftum, in quo author non imme-snor omnibus in rebus eflemodum, nihilnbsp;immifcuit quod alienum , nihil præter-mifit quod neceHarium efle uidebatur,nbsp;Nam perluftratis uariorum authorum,nbsp;qui de proprietate ac numeri difciplinanbsp;traAauerunt libris , quaccuncp ad hancnbsp;artem attinebant collegit, atcp colleAanbsp;( quod in tradendis doArinis optimumnbsp;efle creditur )hoc ordine digeiTic ? PrimSnbsp;contemplatiuam numeri partem fuis fpcnbsp;ciebus annexis artificiofe deducit, fubnbsp;qua numcrum ad Geometricas figurasnbsp;pertinentem compIeAitur, Hine nume-rorum praxim contentasep fub ipfa fpe-cies pratferibit, in quibus certe feitu dignbsp;niflima explicat. Deinde horum omniönbsp;prae'

Præfatio.

piTCdiCtorn fradiones feu partes eleganti docetbrcuitatc. Mox Aftronomicæ fe-quuntur fraAiones-SextS locum abacusnbsp;quàm menfam calculaton'am quibufdamnbsp;lincis diftinclam di'cerc pofTumus, addi-tïs,quocp fuis fpeciebus occupat-Pofthgcnbsp;Tcgula aurea, quam Dctri uulgo ôô corrupte appeUitant traditur. Poftrcmo prgnbsp;ter utiliffîma multa, iam non enumeratanbsp;in cake buius libri inuenttonem tum utinbsp;lem tum uaidc neceflariam Cycli Solarisnbsp;îndidionis, ÖÓ Aurei numeri,quo nemonbsp;quic^ defyderarepofTet, annexait* Porto quid dicam de huius fcienu'ç laudibusnbsp;quàm nemo haAenus improbauit QL o«nbsp;mne feriptorum genus magno huius encomium celebret honored Vereor ne ali-quis idemmihi obtjciat,quod RhetorSnbsp;quidam Herculis laudes cnarrare uolensnbsp;gt;b AntalcidaLacedgmoniorum duceaiinbsp;dire coaAus eft . Quis ilium uitupcrati'nbsp;Siautem id magnipendere uoluerimu»,

A iiti

Præfatio.

quod omnibus negocijs, contradibus Si offîcifSjUniuerîà'^ rerumierieprgbet cetnbsp;ritudincm ac crrorcm qui ex immenfa nanbsp;tare uarietate interdum folet irrepere nonbsp;iîhs eximit animis laudatiflîmam pro-fcdlo hanc artcmexperiemur,quaî hæcnbsp;omnia ex confeiTo in fe compleAicur: ni-hîl enim æqiie rcruin infinitati eftannexnbsp;umac ipfe numerus. Cuius rei Boëtiusnbsp;admirabilis ingentj PhilófophuS;,amp; innbsp;ueftigândis naturæ myfterijs incredibilinbsp;preditus folertia^preciarum nobis adfercnbsp;teftimoniumjquod hoc loco referre mihinbsp;rilacuitjucapudteeofadilius mihifit filles, fimulcp hoc nobile ftudium obuijsnbsp;( quod aiunt )ulnis ampIeAi cures.

' !nquit)cunAis prior eft,no modo quod hanc file hiuus mundanæ molis conditofnbsp;î 3eus primam fue habuit ratiocinationisnbsp;c.v'emplar ,ôd ad hanc cunAa conftituicnbsp;i iuxeunq^ fabricante cocordiam: fed hocnbsp;prior Arithmedca declarat ,quodnbsp;qtix-

PræfatîO»

qnæciincp «attira priorafitttt hisRiblafis ' fimul poften'ora coiluntur* Haec itacp dïf^nbsp;ciplina^quàm ritimeialem fcientia quodnbsp;numcrorum fuppiitandiq^ rationcm pre-fcrïbat,appellate licet,tanto commen-dabilior tibi clTe debet, quanto priiden-florae cautior in tuis rebus agendis cu-pias uidcri. Qiiam in hpe præfenti operenbsp;ftudiofe collects omnib9^ fuis numerisnbsp;abfolutâ cs babiturus. Ex quo, Leótor a-mice,depræhendisauthoris tu diligentianbsp;tu, qua erga te habet bencuolentiâ,hincnbsp;quia iyncero animo hæc Arithmeticesnbsp;præeepta humanis rebus Diuo Hycrontnbsp;mo ad Paulinum presbyters telle ucilif-hma tibi comunicare uoluit, illinc quianbsp;ad picnam huius operis tradudlionemnbsp;cunda follicitiexpreflit, Qiionominenbsp;amicilîîme Ledor te adhorcor, ut hocnbsp;Iludium quod tam innumera fecum ad-fert commoda ftudiofe ampledaris, pa-riterac pronam beneficieadi uoluntatc

A V

Üquo animo fufcïpias,V aie 6d Mulas cx-' ofculate. FriburgiBrtfgoiasnbsp;quinto CalendasSeptero-bris'Anno à Virginia

pareu, « t » lt;r«

Autoris Dodccafh’chon,

Pythagoram noriint omhcs cui gloria princcps In numeris magna non fine laudc fuit»nbsp;Hunc dixiflè feruntillum reAe omniafciie

Pracferric^ altjs,qui numerare fciat»

IdSami) didum,ueio quia çertius ipfo. Incertum ncmo:nemo uocet dubium»

Quanta etenimuirtus Só quanta potentia Aritb-Danda fit,hoe tenui tradidit clogio. nbsp;nbsp;nbsp;( mis

Hanc etiam paucis libuit defcriberc chartis

Atqi adeo forma pingere ritefua.

Naturas igitur nunierorum( candide Lector) Pcnfices:amp;^ quacfo,dexter adcffeuelis,

TYPOGRÄPHVS LECTORI.

T Ailgaris Typographonim qucrcla eft candide Lc(4or,hbros non eadPnbsp;ligenn'a rcuolui à Ledlonbusqua excu^nbsp;duntur. Tametfi enïm uulgo Typogra^nbsp;phi aûariciæ mfiniulcntur,eamcp obcaiJnbsp;fam tantum jtalcm cicdantur adhibercnbsp;dib'gentiam : plcricp tarnen ledî-onunnbsp;fpedant magi's cominodum,quàm pio'nbsp;pnum, Atep ob hanc caufam^ impcdirif'nbsp;fimum quandoep ingrcdi non rcfugiuntnbsp;labyrinthum, modo quam plurimis pronbsp;fint, quodin hoclibello fecimus,jnquonbsp;excudêdo plus molcftiæ expert! fumiisnbsp;quàm cômodiexpcdcmus, propter ua^nbsp;rias Iincarum formulas,numcris tum dPnbsp;ftingucndis turn fegregâdis autalïoquinbsp;notandisparandasjpræterquotidiananinbsp;artis Typographicæ praxim, quod fcrcnbsp;in mathematicïs côringit.QiiamobreiT»nbsp;tuuufmodi Iibros carius diftralicrccogi'nbsp;niü»'

Prxfatio.

mur, æquum eft cnim boui trîttrranti pa* bulâpiæbeatur. Qiiæmûleftiæparuirtnbsp;incaufa fuci unt, curnonnulla non tarnnbsp;caftigatequâ udimùs, parttm cnarnq»nbsp;autor hbelb,procul habi'tSs â nobi's pairnbsp;cas quafdam caftigationes feriusnnlcn'tnbsp;quam expungt potucrint, nempechar-tis ïam cxcufjs. Qtias tamen inmcnda-rum fcricm morecôfueto redegfflemus,nbsp;nifi tain paucæet eiufmodifuiftentquacnbsp;üel a fciolis £lt; Arithmctices tyronibusnbsp;facile emendai î polTcnt ,et nos anguftianbsp;tempons propterinftâtcs nundmas itanbsp;conftiicfti fuiflcmuss.ut exeufa reuoluerenbsp;honuacaret. Hanc admomtiunculam tinbsp;bi candide letftor, indicis mcndarûloconbsp;fcnpfiinus, ut ad ferendum laboi is non-nihil nobifc3,tehortarcmur. Quidenînbsp;Opus îjs premanfum ci'bum in os ingéré-rc,qui probe dentati, per fequeunt inan-dereî' I tacß noftrum hune quantul ûcuncpnbsp;laborc

Præfatio.

ïabóremboni confubto,atcç libcllutti hunc,tibiquammihi commodiorc pratnbsp;do laborious longe minori libcraliternbsp;cmito, amp;alios ubi frudum guftaris adnbsp;cmendumhortator.Quod lîfeceiïsnbsp;alias cmaculatiorê fauente Deonbsp;cundem excudemus, atcp innbsp;hancfpem linear« ÔC tabular« fonnulas feruabimusnbsp;Vale,Calendis Septem-bris,Anni « y j 6.

Ar/thmctieescpitomcliberI: I DE NVMERI DEFI-NITIONE .nbsp;CAPVT.I.

tftquantitasdïfcretorû collediua. vel vt Boctius ait. Eft mul titudo ex vnitatinbsp;bus ag^cgata. Ex rjs fequitur vmtatemnbsp;nonefic numcixim. Id quod alqsquocpnbsp;tationibus oftendi poteft. V t omnis numerus femel infe ducflus, ah'um,pducicnbsp;Vnitas autem femel infe du(5Ia, aliumnbsp;nonproducit.Ergoamp;Tcrltem omnis nu-meri pars eft vnitas, Vnitatisauté parsnbsp;vnitas non eft. Vnitas ergo numerus no

Ad ea quae in hac An'thm eticespart e traiftantur, apertius intelligenda, voccînbsp;quædam dcçlarandæ funt,vt

Naturalis numerorum fériés dicitur, in qua fecundum vnitatis adiedioncntnbsp;fit eorum dedudio.

Arithmctîcei

Differentia numerorum, eff numenw« quo mai'or minorem fuperàt

Numeri â fe aut ab alijs æquidiftât, cîl eorundem æquales funt differentiae.

Numerus peraliummultiplicatur, qu/ toties in vno repetitur, quoties vnita«nbsp;eft in altero, Quic^ cx iftac mul tiplicarinbsp;onefitjprodudus appcllatur.

Numerus alium numerare dicitur, qui inalium dult;ffus,eundem producit. Du-cere ergo eft multiplicare.

Pars, eft numerus numeri, minor quP demmaioris.

Denominans, eft numerus iuxtaqu? fumitur pars,in fuo toto.

Similes dicuntur parteis,quae ab code denominantur num ero.

Oftinis numeri pars eft vnitas PROPRIET ATES.

, Omnis numerus, eft medietas duorS proxime vtrinep pofitorum, coniunc,nbsp;tonnn.vt, 134

Epitome Litîl n

Omnis præterea nûenis, eft medietas duorum vtrincp pofitorum ÄT-Tqualiternbsp;ab eo diftaatium pariter amp; coniuniQo^nbsp;Vt. 4Ó 8

DE PRTMÄ NVMERÎ DIVISIONE. CÄP:ir.

fuiditur numerus primo in parc Ôt^î.

parem. Par cft,( vt Placcntinus definit ) lt;}ui in duo æqualia diuidi potefr,vnitatenbsp;tïicdianonintcrueniente.vcl cft, (vt Pynbsp;thagoras ait )qui cadem partitioe in manbsp;3cimaminimaqï dirimitur. Imparex opnbsp;pofito definitur. INVENTIOnbsp;Præfcriptis naturali fcrienumeris,panbsp;ï’es Sc impares altcrnis vicibus deduci ncnbsp;cefTceft.vt, 1 x j4j65Ccvndehuiufmodinbsp;J5prietas ponitur,Si numeroRr ab vnitanbsp;ïe ^porcionaliü fccudus ab vnitatefucnbsp;*it par reliquos omnes pares cflcsfi Ïpar,nbsp;^catteros imparcs eiTencccfîc eft. vtnbsp;^463 IO

' nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;3^79^^

Arithmetiecs

Tmpares numero, pares coniucfli, par? producunt. numero autem impares, imnbsp;parcm.vt,

Epitome Lib:I. , TIT.

Impar m imparem ducflus fe producif, hoc eft vt lordanus di'ci't, Impar imparcnbsp;numerans, fecundS imparem numerate

Inter parem Stfimparé, vel nullus, vel duo mediant,ft duo vnus par ÔC altenntnbsp;parerit.vt

St par in duas fecatur parteis æquaîcf, vna par fiien'c altera quocp par erit,finbsp;*nipar;amp; altera impar.vt,

An'thmetices

f Panter |5ar Parit ïpar.nbsp;par dupliccs habet jipan'f par,nbsp;fpecicsquarumv^^pTrfe^us,

ru^ t’ ¦ ¦ '

DE Pan'tcr pari,

- CÄPVT: ni.

Ariter par,eft nûcnis par,cuius pai* tes æquai nim fedioncm ad vm'tatë vfcfnbsp;admittunt. vel fecundum lordanû, Pagt;nbsp;nterpareft, que nullus impaniumerat,nbsp;praeter vnitatem. Hanc autetn daufulînbsp;( fdIiçctprætervnitat?)Cafpar Lachsnbsp;adqcit. INVEntio ex proprietatc,

Omnis panter par, fumitur ex ordîe dupliciûah vno côtinue fûptoi^,i'taenî

femper

Epitome Lfb:I nil. fcmper prccedens in binarium ducius fenbsp;quentem producet. vt,nbsp;iq I I 4 8 vel I 4 S »6 Ji] S*nbsp;.ci-T—----—r—rzT -1———r 2«

i i i i 'P^ritj i i i 1

E yri 4 T[i ói pajg| s ,1ÓI jiy64|

Proprietatcs,

Quælibct panter paris pars, nomine quantitate par eft, Nomine, quiade-riominationem habet à pariter pari.nbsp;Quantitate, quod ea ipfanumerus fitpànbsp;riterpar,

Pariter pares ab vno, adnumeratavni* täte, coniunlt;fti,fequentê minus vno connbsp;ftituunt, Vndc ÔC omnes dimiinrti funt.

pariter par ium continue difpofitorum n feries im par eft, ducantur extrema innbsp;fe,0dproduftum æquabitur nontamepnbsp;medio infe du(fto,fed dC citcü^oïitis vPnbsp;adfcrieifincm.vtnbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;B 4

Arithmeticcs

« nbsp;nbsp;nbsp;*nbsp;nbsp;nbsp;nbsp;4nbsp;nbsp;nbsp;nbsp;8nbsp;nbsp;nbsp;116

Si vero feries par fiicrit prodult;flum ex tl emorum æquabitur duobus mçdijs irtnbsp;fe duciis.ôô deinde caeteris,qui medijs alnbsp;trinfccus adijduntur,ad fincm vf^ feri'nbsp;d.vt.

CAP.

Epitome Libîl V» CAPVT nil.

P Anter impar eft numerus parcuiuî media partioncm xqualium non admit

INVENTIO.

Panter impares fiuntex imparibusab Vnitate naturaliter fumptis,in quos ft binbsp;narius-ducitur.vt.

Omnis numerus, cuius medietas eft impar,pariter impareft. .

Paritcr imparis partes quantitate lt;Jenominatione difcrcpant.Nam fi quinbsp;titaseftpar,denominatio erit impar, 8Cnbsp;lt; contra.

Inter continuos duos SCproximos pa-Bj

’ Arithmcticcs ' rttcrimpares,tres nueri naturalitcrdif¹nbsp;pofiti mediant, vt.

¹

I 6 |ioîi4'»8jxa!x6'59|34

Omncs item pariter impares quatema riofeexcedunt,vtpraeccdenti exemplonbsp;vides. Vnde differcntias eorum^æquale^nbsp;cflcnccefteeft.

Pariter impariura pari SCnaturali fcrft difpofitorum.duomcdij coniuncftifuisnbsp;numeris vtrincç ad vnitatS vfcppofitisnbsp;Ä colledu «quantur, vt,

€

Arithmetices

DE IMPÄRITER PARI CAPVTV.

JlMparitcrpar eft numerus par,cufus acqualiumfecflio, nonadvnitatem vfcpnbsp;peruenit»Vnde lordanus Impanterparnbsp;Ônquit) eft,quæm quidam par fecundiînbsp;parem, SC quidam, fecundurnimparemnbsp;complet.

INVENTIO

IMP Ariter par 6C panter pari SC panter impari adfimulatur. Nam vtriufc^ vices gerit.SCproindc quum panter parnbsp;à paribus abvnitate duplatis fiat. Paritnbsp;impar autem ab imparibus â temario ofnbsp;tum ducentibus, hune quoep numeruirtnbsp;ex vtro^ fieri conuenit.Impares igitUfnbsp;â ternario deducli SC per panter pares, ânbsp;quaternario deferiptis multiplicati,nbsp;ducunt Impariter pares .vt *nbsp;4 8 16nbsp;nbsp;nbsp;646CC,

! 7 9 * « SCc,

Omm's impariter pan's parteis quaedï ^cnomi'nanóe ÔCquantitateconucniQtnbsp;S^ædam verodifcrqjant.vt IX habet binbsp;barium partem quantitate 5C dénommanbsp;• ï’ona parem. Dcnominatur eni'm â parinbsp;’f’6 quia binariusfexta pars cft i xDenonbsp;nu'nati'oneprtètereapar eft, quia 6 eftfcnbsp;lt;unda pars fiue medietas i z. Porroidénbsp;Humérus i xhabet temanum parte quan

Anthmrtfcc» ' titate imparcm,fcd denominat/onc pa*nbsp;rcm.cftenimquartapars i x.

Numcnis abinan'o non duplus, cuiirs mcdi'etas par,impan'ter par eft,vt__

1 x’xo[ *4 [Impariterparcs.

1 X

'ë

_

Ex dudlupariter pan's m impariter p» rem,quot;

Ïmpanter pares.

Panter pares. -ïmpanter pares, eX ‘ ducT^upnmi panï pa^nbsp;n's in impanf parem.

ïmpanter pares eX duc?Iufecüdi pan? panbsp;n's mimpant parem.

ïmpanter pares, cX dudu tertq pan'f pa*nbsp;n's inimparitparem '

faciunt lio

Si panter impares numero pares coa-^^ruentur, compofitus erit vel pariter B^jVelimpariterpar. vt

DE PERFECTO

CAPVT VI

^Junc de cætcris pan's numen' fpettC' bus dicendû eft. ôCprimo de pcrfedîo.

Pcrfccflus igitureft numerus par, cU' fus parteis omnes conïundlæ fummarunbsp;totius præcife conftituunt. Pars hoc lo'nbsp;CO eft, quæ aliquoties fumpta, totum tnnbsp;vnguem metitur,

INVENTÎO

Panter pares ab vnxtate natural i {cric

Epitome Lib. I

dcfcriptiper additionem col liganturamp;^fi in viium itaconnbsp;gcfti, numcrum pnmum conbsp;ftituerint, ineûdêprimû feznbsp;incompofitS ducaturcol-^Ptflorum maximus èC in pronbsp;^uclo perfedus apparebit.

vt

Än'rhmctices

Sunt aut em perfcdi admodum paud in monadicis enimfolus, eft .6 Jn dccx'nbsp;dicis X 8.Inhecatondicis.4p6.inchil/a-'nbsp;dibus,81Xs.Etqufntuseft.

Proprictatcs,

Pcifeöus alternatim iâi'nfenarium# fam in ocflonarium definit.

cæteræ, fi quæ funt ,pprietates ex dimi' nuto Stfuperfluo dependent.

DeDiminuto Si fuperfluo, Caput vn.

32)lminutus eft numerus par quidcm, cuius tarnen partes minus toto ftatuuntnbsp;Hic amp;:impci-fclt;ftus dicitur.

Superfluus eft numerus parcuius par' tes coacfæ fummam totius cxccdunr.nbsp;Vocaruretiam abundans.

Diniinutiamp;^ fupeiflui multi varijtp fûtjacf’nevllaordinisobfeiuationediftnbsp;perfi, Vnde inuentio eorum incertanbsp;eft

Epitome nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;X

eft Ä^vagabunda. Eft tarnen ah’qua «gt; •cniendi ratio.

Inucntio Diminuti,

Arithmetics

Abundans comodiffimepcr .60. me furatur.Omnes entra hums numcripat'nbsp;teis(quæ ÔCipfæ numerorum cenfentiWnbsp;nomme}abundantes funt.

Proprietäres.

Quencunqj perfedus, aut abundan» numeratjidem quocp abundat,

Omnis perfedum numeraiis,cft dim* nutus.

DE Numero impari. Caput VIIL

umcTUS impar eft, qui m duo aequa' lianon poteft diuidi'.Inuenn'oncm Só,pnbsp;prier ates quære fupra de prima numcrinbsp;dmifione.

Impan's tres numerantnrfpecies. Primus SecundnsöC ad alter« primusnbsp;DE Primo ÔC fctundo.

Caput IX.

Primus

Epitome L/b. I. XI. r^Rimusnumenis cfl:,quemfolx me-ti'tur vmtas.Hic alio nomine dicnur In-compofitus. Qgt; fi duovelplures Inco-pofiti inter fe comparantur,Cótrafeprinbsp;mosnominant.vt ? fiC ƒ.

Pono numcnis numerum mctni dici-tur,quum vel femel,vcl bis,vel ter,vel ties veil's,numerus numero ccmparatusnbsp;cundcm totum præcife conftituit.

Numerus fccûdus eft quem præter v-tu tat cm, al ius menfurat. C ompofi tus a-has vocatur.Si hums generis plurcsfunt Commenfurabiles feu communicantesnbsp;»ppellantur, vt,9

InuentioPn'miex propnetatc.

Omnis numerus Primus aliquisi'm-parium eft I'ta deduCtorum, vt qui peft tiullumimparem aut aliquem fupraip-fum,totusvcniat, quotus aliquisimpa-tium fuerit ab vnitate,vt. $ primus eft,nbsp;C ?

Afithmctices

fed non to tus poft aliqucm imparnrm* quotimpar illcab vnitatccft. Nâ $ eftnbsp;primus poft 3 innaturàlinumeroru ferinbsp;e.At j^tertiusab vnitatc. lté 7 eft fc'nbsp;cundus a j qut tenius eft ab vnttatc.Nonbsp;eft igi'tur idem ordo pn'mi ad imparcm,nbsp;amp;imparisadvnitate, Quotus 06totusnbsp;VtTcrtius.Quintus Septimus, vt.

Inuentio Compofiti.

Omnis numerus compofitus poft aft quern imparlum naturah fcrie difpofitanbsp;rStotus eft,quotus numerus illeimpar»nbsp;ab vnitateeft,autpoft aliquem fupraipnbsp;fum imparcm totorum totus,vt difpO'nbsp;nantur impales naturaliferie, itanbsp;»jjirpiiijijiyip Sgt;Cct lamnbsp;tcinai ius eft ab vnitate fertius, fumatuf

Epitome Lib;I, XII

ergotertiusfn ïmpa' Ç Mu feriepofttoman 31 3nbsp;Umexclufiuc, nem-pe . 9. Hicergo pernbsp;pn'mam partem hu-ius proprietatis eftnbsp;Gompofitus, terti-Uscnimeft poftim'nbsp;parfum alfquem, vt-potepoft temanu.

Omnis numerU' primus,ad quern nonbsp;ftuitierat,primus eft

Omniscompofiquot; rils aprimo numera-

vt, C4

Impoflibile eft duobus contra fepn'mi« tertium in continua Proportionalität«nbsp;applicare.

De numero Adalterum primo.

T Caput X,

Ertiafpecics Imparis eft numerus adalterum primus,Hicperfe quidé Sc'nbsp;cundus eft èc compofitus: adalterû veronbsp;fi comparetur,primus amp; incompofitusnbsp;cftvt 9 ad 1 ôjQuia 9 temario ter fumpquot;nbsp;tomenfuratur.at lôternarius nonum«'nbsp;rat .i.aliquoties fûptus non c ôftituitjit»nbsp;huius generis numeri,quia praeter vn»nbsp;tatemnon habcntaliam paitemnumC'nbsp;rantem contra fe prim i dicuntur.Num«nbsp;riveronumerantem habentes vocantufnbsp;commûicantesfiuccommenfurabile*.nbsp;vtpôCia.nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;f

Ärithmcticcs

Hacîlcnus dcfimplicinumcrr confî Je* ratione didum; Nunc de numero age^nbsp;musrcfpeÆ'uo, Numerus igiturrelatênbsp;uc.i.ad ah'quid çonfideratur fccunduntnbsp;sequalitatem Jnæquah'tatem. QuiC'nbsp;Huid eni'm in comparationem Si refpeC'nbsp;tumvenit,autfecundum acquale fit auCnbsp;ïnrcqualc.quoru illud fcmpcr vno mo*nbsp;do, hoc autfecundummaius SC niinugt;nbsp;contingit.

Duorum porronumcrorum refpcct3, fecundum maius qui fit, Inæqualitatctnnbsp;vocant maiorcm. Alters vero minoren»nbsp;inxqualitatem.

•Eftcp inæqualitas maf or, quan do ni* menas maiorad minorem confertur. vt,nbsp;4 ad i Minor iuîcqualitas vt x ad 4,nbsp;De Speciebus Tnæqualitatis.

Caput xi

Aioris in^qualitatis fpccies numlt;

rantuï

Epitome Lfb;T XITII»

tantiir quincp.vt multiplex. Superpar-ticulariSjSuperpartiensjMukiplcx Su perparticularis amp; Multiplex Superpa»nbsp;tiens.

minore aliquotics praccife continet, vc bis, ter, quater, Huius fpecics infini-tæ funt, nam fecundum quodminorcmnbsp;Varie continet,nomcnquocp variât, vcnbsp;fi minorcm bis habuerit nominabiturnbsp;tiuplus.fiter, Triplus. fiquater,Qua-«lruplus,ôi^,c.

Inuentio Dupli.

Præfcribantur binario pares, qiri

Arithmetic« Inuentio tripli..

Præfcnbantur à ternanonumeri holt; modo vtpoftfingulos binarïus inter-mittatuf.ad quos deinde numeri ab vni'nbsp;täte continui confcrantur. vt.

Inuentio Quadrupli SCaliorum.

Pingantur a quatcmario numeri fit:» vtpoft fingulos ternarius negligatur»nbsp;Ad pofteanuméros ab vnitatc conU*nbsp;nuos referas, vt,

Eadem proportioncquot- ~8~J~ quot multiplicis fpecies ha- ——nbsp;bcrelibucritjinucnies. I-* %

Epitome Lib. I, nbsp;nbsp;nbsp;XV.

Ad omncm inæqualitatis fpetiem re-pracfcntandam duonumcri funt neccflx «Î.

Omnisnumenis, ad vnitatem fi refera ^ur,fpeties eft multiplicis.

Si duomultiplices eiufdem fpecieicon ïuiufti fuerintjCÓpofitus erft multiplex:

De Superparticulari. Caput. XIII,

Q

O Vpcrparticulan’sf fccunda maiorum inæqualitatis fpecics)cft numerustotSnbsp;fibi comp^iratum amp;aliquotam comparanbsp;tipartem in fehabens,vt,4ad 3. Eftautcnbsp;pars aliquota numcri, quæ aliquoties acnbsp;^eptatotumpræcifc conftituit, vt, gt;adnbsp;*jNam ternarius binarium non folumnbsp;^otum habet, fed eiufdem di'midium.nbsp;Ïta4 ad 3 gt;hoc eft.Quaternariusteinari-tium totum SC tertiam eiufdem t ernarij

Arithmet/cee

pan cm, quxvni'tasSC tertia eft, conti' net»

Inuentio Supcrparticukris,

Superparticularcs nafeuntur. Si praß fcriptîs numen's abinario continuis, fc'nbsp;quens ad liïiediate precedentem comp*nbsp;retur. vt,

î Sefquialter.

2-,' Scfqin'tcrtius. Scfquiquartus.nbsp;Sefquiquintus.

A] Sefquifextus.

r nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;¦

8

Inuentio Sefquïalteri.

Numeri a temario duobus fempet poft quæmlibet, mtennilîîs continüinbsp;quos

Epitome. Lib: I. XVI (quos nonnuJli Triplos vocant ) ad pates abinario natural iter præfcrfptos c®nbsp;paiati Scfquialtcrosproducunt. vt.

À

InuentioSefquitcrtq Sfraliomm

Simili modo nûcri o. quaternario tri-^usfemperomiflis continui comparati numéros a rematio, duobus fem pernbsp;*'^§ledis,proccdentis,Sefquitertios conbsp;^Uuunt.Eadem ddnde proportiôcnu-^cronim obferuata quotquo t vol ucris,nbsp;^uperparticularisfpecies inuenire lice'nbsp;Vt,

A nehm et/ces

Proprietates:

Omnis fupcrparticulaiis’.mfnorem fubtilioremfe, Superparti cularem poftnbsp;fehabet.Huic proptietati Hemitom'olFnbsp;ratio fubiacet.

Minor autem fupcrparticularis cft» qui âmaiore numero fuam habet app^^'nbsp;lationcm.

Sivniusadaltcrumratio fuerit muh^' plex;tonus ad maiorem proportio en^

Epitome Lib J, nbsp;nbsp;XVII,

Superparticularis.

Sola fupcrparticularium fefquialtera ^ft:quæ cum nulla multiplice, raultiplinbsp;facit fupcipartientem,

Numeri ab vnitate fi pingantur: duo priores Multiplicemtcæteri vero fuper--particularium fpecies coftituent.

Omnis fuperparticularis adiuncîafu-perpartiente: prop or tionera producit tripla minorem,

Diucrfi Superparticulares duocoiunc tijvcl duplam vel fuperparticularenbsp;-tiunt vel fuperpartientem»

J)e Sûperpartientc» Caput XIIII,

Ç

C Vperparncns ( Tertia maioris inae-qualitatis fpecies ) eft, quum numerus maior minorem totura cum aliquot ei-

D

Arithtftetfc«

üfdcm partibus comprachédifjVtp ad 7quot; Suntautem fupcrparacntis partes nöl'nbsp;abquotæjVtin Superparticulari, inhoCnbsp;enim partes funt vt medietas, Tertia»nbsp;Quarta,Sc c. At in illo partes, vt Duæ»nbsp;Tres, Quatuor, Scc. Huiufmodiigitittnbsp;partes in Superpartientefunt, quatMi'nbsp;noris partem aliquotam non effîciunt»nbsp;Denominatur cnim fuperticns â name'nbsp;10 partium numeri minoris, quar vltr» 'nbsp;ipfum in maiorc continentur, vt Maiotnbsp;Winorcm totum habens 6C Dua^nbsp;ipfius partes vocatur SuperbipartienSnbsp;vt ƒ ad ? vel/ad j,Prætertotum aut^nbsp;fi tres Minoris partes in Maiorefuerin*nbsp;Nominctur Supertripartiens,

Inuentio Superpartientis,

Numerià temario continui comp*'

Epitome Lfb:I XVIII

*^tiadimpares aquinan'o cotihuoSjSu* perpart j'entes conftituunt.vt,nbsp;-------? j I J Superbipartientes.nbsp;7|4 Supetnpartientes.

’ »1 6 I Supcrquintiparneres ’^Irt Supcrfextipartiétes.nbsp;«^yj 8 I Supcrfepnparhctes.

Inucntio Superbipartieutis ÔC aliaium fpcderum.

Sopn-bipartientes Hunt: fi in primi fuperbipartientis numéros ducatui bi-Harius,vt bis funt i o bis j fut 6 Pcftcanbsp;in producftum illud,quod fecundum fu-• perbiparticntcm iam indicat idem binanbsp;tiusduóus tercnim fuperbipartientemnbsp;producit. Ita quoque binarius inpro-xime produdfos termines diiotus,nbsp;ahum proximum Superbipartienten»

Arithmcticcs

procréât. Simili modo ternarius multi' plicatus per primum Supertripartientcnbsp;producit fecundum Supertripartieiitç,nbsp;îtemcç ternarius in fecundum Supertri'nbsp;panientcm duôius,conftituit tcrtiû Sænbsp;pertripartientcm Qéc, Ita Quaternari-us in Superquadripajtientê duc^us, Su'nbsp;perquadripartientem facit QCc.'vïj

Epitome. Lib.I. XIX

¦ Numeranturpotius denominantur partes fupcrpartientis.

Superpaiticns fi eoraponatun adma^ lorem ent fupcrpartiens: adMinoremnbsp;Vero Multiplex fupeipartiens.

Quilibet duo fupçrpartientes eoniSc-ti proportionem quadrupla minorent conftituunt.

Quiuisfuperpartietes in fuperparticu lares redudpoflunt. *

De Multiplici fupet' particulaii.

¦» nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Caput XV.

JVY Vltiplex fupciparticularis ( quar-ta maioris inatqualitan's fpecies )eftca Humerus maior minorcm aliquon'es in-cludi't cum ciufde aliquota parte, v t mi-norem bis cum fua me dictate continensnbsp;eft Duplus fcfquialtcr. SiipfumbiscSnbsp;tertia, vocaturDuplusfefquitertius. Sinbsp;ter cum tertia, nominaturTriplusfefq-tertius

Arithmctices

tertîas.S^c Et fîcfpeciesmultiplias fu* perparticularis ex multiplia fuper-particular! SC fupcrparticulan’s aliquo'nbsp;ta parte ïn infinïtum extendi poflunt.

Inaentio multiplias fuper* particularis

Ad iraparcs a quinario fignatos adap tcnturnumcri abinario null o intermix

A. feptenaro fciibantur numerf duo* busfemper intermiflîs ad quosnumcïlnbsp;binarium fequentesadplicentur. vt.

T ripliruper* particulares

Quadnî»

Epitome Lib. I XX

Qviadrapli Supcrparticulares mue-tu'untur. fxaNouenario, tribus fempcr negleólisj numcri præfcnbantur, quitus numeri * binario poiiti adcomodarinbsp;debent. Quincupli fiunt fl ab vndena-rio, quatuor obmiflis, numeremus: SCnbsp;numéros applicemus a bihario fignatosnbsp;Simili modo luxta proportionem nu-*nerorum mtermittcndorum quotquocnbsp;habere volueriSj multipliccs fupcrpartinbsp;Culares, inuenies. Semperq?flct vt nu-*Hari a binan'o poflti fpcciabus accomo'nbsp;dentur. Honina omnium banc imagi-’

D4

Epitome LiblJ XXI,

Jnnentio fpccicmm Multiplias Superparticularis.

Dupli Sefquialtcri fiunt fi numcri à bi nario pares conferanturad numéros ànbsp;* quinariofcriptoseodemepfe fupcrates,nbsp;vt.A

Tripli a. temario fumpti SC ad feptena* riumnumcrofcp codem fetranfeenden-dentes relati puplos fefquitertios con-ftituunt.vt,Bnbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;' .

Quadruplis â quaternario defcriptis accomodentur numcri â nouenario femnbsp;tgt;cr nouenario maiores,. ôi^fiehtfefqui-9uarti.vt,C

Ex quincuplis 6C vndenarijs nafeuntUT E)upli fefquiquinti .vt,D

Simili numerorum feruata proportion multas alias fjjceicsinijcmreliccbit.

Ep/tomeL/b.I, XXII

Ex paribus â binario 5d feptenarijsa feptenariodigeftisTripli fiunt Sefqui»nbsp;alter/, vt A

Ex Triplis atemario, Ô^Denarijs a denariodifpofitis fiunt tripli fefqu/tex*nbsp;tij. vt, B,

Ex tredecim, QC ex dedcs SCter fc tranfil/ent/bus: adh/b/tis a cpratemar/onbsp;quadruplis fiut Trip! i fefqu/quarti. vtC

Confimib extenfioneperproport/o ftftn fadaplures inqu/runtur fpcdes.

Tripl/

Äritfimetices

Proprtctates.

Sola fupcrparti'culanum fefquiattc-tamultiplicem fuperparticularempro' j duci't multiplex fuperparticularis adiö^ inbsp;gitfimili multiplici, Superparticularê Înbsp;denominatam à numero qui fit ex duC' 'nbsp;tu multiplfcis in partem.

Supeiparticularis ôi. Multiplex fupcr particularis cum eadem MultiplicirprOnbsp;portionesfimiles conftituunt.

Si maioris ad minorem proportie mul tipliciiuiitgitur produótum erit autmiilnbsp;tjplex,aut multiplex fuperparticularis»nbsp;aut multiplex fuperpartiens.

De Multiplici Super-partiente.

Caput XVI.

multipl^^ I

Epitome LibJ. XXIII.

Mvuit î'plcx Swpcrpartiensf quinta maions inacqualitatis fpecies ) eft cumnbsp;tnaior numerus minorem, cui compara-tur,aliquotïes vna cum aliquot ei'ufdcmnbsp;partibus includit.vt 13 ad j Etfanehæcnbsp;fpecies totum â multiplici:parteis autënbsp;8 Superpartientefumit. Et ab vtriufc^nbsp;Variatamultitudine fped'es multiplicïsnbsp;^perpartieti's dcnominâtur.vt, duplusnbsp;Äraplus Supcrbiparo'ensSüpcrinbsp;^tipartïens. Superquatn'partiens.

Ocftonarq ab ocftonario: amp;a T ernan'o ^riplïDuplos Superbipartientes efficinbsp;^t vt.

Anthmctica

Vndenarfj abvndciiario’.amp;fQaatd'' narij a Quatcman'o .pgreiïi Duplolnbsp;fupcrtripartientcs conftituunt. vt.

Dccimiqnarri a QiTatuordecimtamp;fa Qümario dcfcn'pti Quinarfj Dirplos fanbsp;Ciuntfuperquadiiparcientes. vt.

Vndenarij ab vndenarioamp;T aTcma' rio Tcmarrj ftatuunc Triplos fuperbi^nbsp;partientes. vt.

Epitome. Libi I» XXIIIl

A qufndecîm Denarij quiharij ad qua ternarios a Quaternario inchoantcs co-parati Triplos producunt Supertipartinbsp;«ütes.vt.

Itafpecics Multiplias fupcrpart/en produces in infinitum ü fempcr pronbsp;Maiore numerum acceperis, qui proxi-*^efumptum tcmàrio excedat: Pro Minbsp;»^orevcrofi numerum fumpfer s qui^-3tîcf5ptovnitaTemaior lît.itatamc çî

] I [T'. ' ' Z . ? Arithmct/cesl fupcrbipartientftus Minor fempar fit

De Minori inxqualitatcamp;^ ciufdcni fpecïebus»nbsp;Caput xvn.

.^^.Inor inæquabtasjVt paulo-fupcrf' US diximuj, eft curn numerus minofnbsp;«

maiori comparatur, vt i ad 2 0:^ 1 ad j Habet autem cas,quas Maiormæquab'nbsp;tas,fpecies ijfdemquocç nomimbusap'nbsp;pcllatasmfi qjfinguhs prxpofitio Su^nbsp;przfigitur, vt dicâus Submultiplcx fubnbsp;fupt’f

Epitome Lib. I XXV fupcrparticularis ÔC fubmultiplex fub-fupcrpartiens»

Porrofubmul.

tiplicis fpês.ut.

Omnis Ipecies maioris ihæqalitatis, tranfit in minoris inacqualitatis fpeciem,nbsp;fi eidempraefigatur: Subminorq; nume-f US maiori præponatur in exemplis.V n-?ie öd omnium minoris inæqualitads ipe-tierum inuencio,ex maiori petatur æquanbsp;litate.

^ue inæqualitate fadas,proportiones uo-îant.

De numero ad Geometricas figuras pertinente,nbsp;Caput XVIU.

E I

Än'thmetices

J^umcrus Geomctricus figuras fcciiB dum unitates ordinans aut Linearis eftnbsp;aut Planus aut Solidus,

PROPR.

Vnitas omnis, Gcometrici nftmcrf ritimaginem.

DE LINEARI NVMERO. Cap. XIX,

-^Jvmerus Linearis eft qui a binari® fecundum naturalem numcrorS ferie»’*nbsp;cxtenditur, vt 2nbsp;nbsp;nbsp;nbsp;nbsp;6 vel qui fuis puU'

lt;ftis num er um in vnitatem rcfolutü d^ fignantibus fecundiS longitudinem dc* ,nbsp;feribitur VC

I

De Numero plano, ¦ ¦ Cap. XX.

J^Vmerus Planus feu Superficial* eft quiperunitates fuas inlongum, 0^^’nbsp;tuO*

EpitomeLib.I. XXVI. tumtcnditur.vt 3 •nbsp;nbsp;nbsp;nbsp;6 •

• • • •

• • •

Internumerus Pianos alius Trïgonus «ft, alius Quadratus, alius pentagonus,nbsp;allUS Hexagonus, dC fi c fpecierS infini-tas eft.nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;P R O P R.

Omnis numerus Planus ex Trigonis tomponitur.

Semper duo plani proximi adfe col* Uti ( vt Tetragonus trigono, Pentago-nus tetragoiio, amp;c. triangulo fefe tran-ftendunt.

De Trigono. Cap. XXI.

.1 Rigonus eft numerus planus qui v-J'ftatibus fuis n'tcdifpofitis tresangu* totidemque latera , uc Ifopleuros

£ 1 apud

Arithmeticcs

apudGcometraSjhabet.vt •

A.V.‘

Inucntio.

Numéris fecSdum ordinem naturk' lemabvnjtate difpofitis: fequcns ant^'nbsp;uertcntibus adieflus Trigonum conft*nbsp;tuuntvt, JCT

Adhanc Trigonorom mucntioneni baud parum facere videtur Progrcflïenbsp;pradïd numeri fpecies.

ü»

Epitome.Lib: I. XXVH

3

•

• • • •

6 nbsp;nbsp;nbsp;nbsp;•

• • • • • • •

• • ••• •••••

Proprietates,

Quemadmodum vnitas mxmeri 8C x H'ïalitas inæqu^litatiSjita temarius niç-Plani pnncipium eft.

Sià quouis trigono latus fubducatur ’Pparebi't m rcftduo Trigonus proxi-minor,

Omnis Trigonus duplatus: Alter» parte longiorem conftituit,

Trigonus cuiufui« altera parte longi •ris medietas eft.

El

Arithmcticcs

Duo proximi quicp T rigoni confim* ôiQuadratum pracbent.

T rigorii ab unitate fi dcfcribätarjdU' os priores ïtnpares:fequctes duos pares:nbsp;amp; iic al term's mcibus pares ÔC impaiesnbsp;ordinan' eft nccefle.

Si Trigoni poft um'tatcm fignati ftiC' rinf.inter duos 0lt;r duos difiundim acce'nbsp;ptos proportioncscrunt,quæ interna'nbsp;merosab unitate,nullo intermiflb,dc'nbsp;fcriptos.vt J 1

1 o

i, I

x8

De Tetragono, Cap.XXII.

X et ra go ft us feu Quadra tus eft num« ttts Planus qui fecundum fuas vm'tates

Epitome Libîl XXVIIL quatuor an gulos ÔC lacera diftenditujf

Inuentio.

^lem ordinem digeftis : fîfequens antc^ ^edenti adijciatur,verc Quadratum cffî-'iet. NominamusautcmucreQuadra-^Um cuius omnia latera funt acqualia. vc

• Facit amp; ad banc Inuentionc cito pr«gt;-ahendam Aritlimetica Progreflîo. Quadratifuntdiucrfî.Na alij latitudinbsp;ï'c acqualê habet logitudini; ÔC hosvercnbsp;quadrates nôino. Abojjt latera vnîtatis

E 4

An’thmeticcs tantum difFcrétiafunt diuçrfa vt bis trianbsp;funt 6 autTerquatuor i i.Quatcrquinnbsp;«P i o.IIIos altéra partclongiorcs Lôgi'nbsp;laterosvocât.Alij denicÿ funt quorumnbsp;latera plufq vnitate diTcrcpant vt bis qn

Etillosantelongiorcs ant parte Ionquot; giorc»,aut vt valla,Praelogos appellâCrnbsp;Inucnti'oncs aliæ vcrcnbsp;Quadratorum.

Tetragon/ex duobus quibusty prOquot; xtnui Trigonis oriuntur. vt,

»

Epitome Lib. 1. XX DC

Omnis numerus femel infedudàs» i'e Quadruplum producit.

Altera partelógiorcs ab vnitatcfup. proximi quicp duo coniundi fum-*^amprarbent,cum8dimidium vereTc gt;nbsp;^Ägonuseft,

Inucntioncs altera part« longiora.

Digeftis abinario paribus: ß fequens antecedcntibus iungitur Alterii partenbsp;iougioretn conihtuit. vt.

Arithmetics

Digerantur à biriario para : qiribus temario impara ad latus applicentur.nbsp;Deinde pares in imparcs ÔC impar« i»nbsp;para altemamultiplicationc ducantur»nbsp;exaltera parte longiora producêtur.u€

Verc Quadratis à quaterftario difpo' fitis addantur numeri naturales à bina'nbsp;tio deferipti âfproducçntur Altera pargt;nbsp;eebngiorcs.vt

Antelongiorcs omnesfuntquipro-^ucuntur ex multiplicationc numero* *^mquoi-um Maior Minoré plufquaianbsp;Vnitateexcedit.

P R OP R.

Omnes Tetragoni vna iunguntut / fticdietatc.

Quadratus in Quadratum duélös, Quadratum in Summa ponit.

Quadratus autem Altera parte Ion* giorcm multiplicans, Quadratum nonnbsp;producit.

Quadrat!â Quatemario defcriptiSi fubtradiiab Altera parte longioribus ânbsp;Senario digests : ponuntin reilduonu*nbsp;mcros

Arithmetic»

mrros à binanb naturales

DcPcntagono, Hexagon« alijfcp pleni numcrinbsp;fpedebus.nbsp;Caput xxni.

TPcntagonusquinqjificHcxago^ nus fex angulos SC æquahalacera contiquot;nbsp;net. Heptagonus, Odagonus,Hema-gonus,Decagonus,âCc:cxipfavocabUquot;nbsp;li fîgnificatione defciibuntur facile »

Inucntto Pcntagoni.

Tn’gonis ab vnitate digeftis Qua* draiiaQuaternariû defcripti SCaddtti»nbsp;Pentagone« générant, vt.

AI»

Epitome. Lib.I. XXXI Alia.

Digeranturabvnitatenumcri naw-*^cSjamp;poft vnitatem duobus obmiffis fcquenSjVnitati adiedus. Pentagonuntnbsp;Conftituit.Simili modo in fubfequenti-bus duo femper intermittuntur. Qi. fe-quens cum prioribus numcris, qui duobus negledis notati funt, Pentagonsnbsp;ftatuit, vt.

Arithmetics

Inucntio Hex a go ni.

Qircmadmodum numcris ah vnitits fiaturaliterpracfcriptisetpoftjvnitatemnbsp;duobus Temper iiiterceptis fequens cumnbsp;VnitatePentagonumu'tatribuspoftvninbsp;Catem neglelt;fris,Haxagonum coftituit*nbsp;Et ficut fubfequentes Pciitagom per dttnbsp;os-jita per tres numéros interceptos Hc'nbsp;xagoni producuntur continui.ut.

Alia«

EpftomcLib. I, XXXII.

Alia.

Offlm's Hexagonus ex Pentagone ÔC proxime anteuertentc T rigono companbsp;ftitur.vt,

Alia.

Trigonisab vnitate digcftis, tertius Hexagonum oftendit,iquo deindetcr-tills alium Hexagonum . Etficfempernbsp;ib Hexagono fequens tertius T rigonusnbsp;fequentêtê ponitHexagonû. Vndema-hifeftûeft quod omnis Hexagonus eftnbsp;Trigonus.vt,

Anthmetices

ïnuenfio Hqsfagonî.

Heptagonus ex Hexagono 8^ gono conflatur.vt, ,

EpitomeLib.I. XXXIII

In Hexagone conftituendo prefen-ptis ab vnitatcnumcMs naturaliter inter tnjttunturtres. AtinHcptagonoinue-hiendoobmittunturquatuornumeriJnnbsp;octogano quincp. InHennagono fex.nbsp;t)einfimili numciom obmitcendónini

Ari thm Ct ices De numero foil donbsp;CapJQOIUnbsp;j^\Jumenis folidus eft, qui per fuasnbsp;nbsp;nbsp;nbsp;'

tates digeftus, longttiidmi SC latitudi^ craffitiem fuperaddit,hoc eft,trxno diü^'nbsp;diturinteruallo.

Porro numeroniminaltumpofitofgt;^ diuerfe func bafesiah'oriim eteni'm cri3lt;^'nbsp;gulæ,aliorum tetragonæ, ô^c.Et horui^nbsp;omnium quidam lacera habent compilénbsp;ta,ac dtcütur pyramides. Qiiidam hab^^nbsp;latera ufqueadfummum fine conuni n*’nbsp;conuenientia, SC curtæ pyramides fünf»nbsp;Qindam habent latera æqualitcrS^fn’’'’nbsp;rcdla Sd diftantiazhique fuperïorem SCnbsp;feriorem fuperh'ci'cm habent æqualeif’nbsp;è quibus, bafis trigona fi fucrit, fcrralt;fl^‘nbsp;les funt. Si fecundum bafim quadrang^’'nbsp;lam in omnes dimcnfîoncs extendantU^nbsp;«qualiccr,cubi uocantur. Habentesai^'nbsp;temahef* '

Epitome Lib.1. nbsp;nbsp;nbsp;XXXIIIL

, fem latera æquidifiancer eretSaß^bafes pcntagonos plurimumuè angiilorumapnbsp;fiellantiir coîumnæ^Denîqiie quidam di»nbsp;. mcnfiones omnesnonexæquodiftribu-^nt, quorumjalij dicantur Laterculi,alijnbsp;AflercSjalij Cunci,alîj Parallelepipedi«nbsp;Bafis eft 1 inca iacens.Conus eft furrenbsp;^icorporis fummitas’.ô^innumcrofo'nbsp;ïido vertex eft èC vnitas.

De pyramide. Cap.XXV.

yrimls numerus folidus eft anïis latera ab aliquo numero plano ad fummu leuantur.Et hæc a Trigono Trigo-t^agt;TctragonaâTetragono, 8Cc. deno-ttiinatur.vt.

Epitome. Lib.I. XXXV.

Trigo na f’yramidis ergo fpccics font Quadrata

Pentagona

Pyramis perfeda eft, cuius latera ad Wnitatcm ufque leuantur.

Pyramis imperfc0:a cuius laterum et ’fedlio conum,ut unitatem non attingitnbsp;Et alia eft Curta alia Bifcurta alia Tricn*

Pyramis Curta eft,cu i in ef c Aione mo tias deeft.Bifcurta, quæ monade amp; planbsp;no numero unitati proximo defticuitur.nbsp;Tricurta eft,cuicum mon dae duo plantnbsp;defunt,Et fie de cæteris ,ut

Arithmcticcs PROPR.

Pyramidum dcnominatxo cxplano numero eft.

Omnis numeri folidi pn'ncipium Py ramïseft:

CuiidwsPyramidis bafis numerus pla norum maximus eft.

Quotlibct T rigom' acquales in altuiö Compofiti.Serratilem producunr,

Omnis Serranh's Pyramide fuae ba' fis duobus altioneidcm triplus cflepro*nbsp;batur.

DeLatcrcuIis. Cap. XXVI.

T . A terculus numerus fotidus eft qui ft cundumfuas defcriptus vnitateslongi'nbsp;tudincm aequatUtitudini: cöncifioremnbsp;tarnenaltitudinemhabet,vt i s.

Nam ter triafunt 9. Et bis noucm funt

Epitome Lib. I. nbsp;XXXV7lt;.

J longitude»

Ita 3 latitudo

X Altitudo PR OPR. Alntudo laterculi caetcris dimêûonîb*nbsp;''nitateminorexiftes,arquipollet Alte»nbsp;fa parte lôgioruQuot fi plus vni'tatemjnbsp;îiOifiicjitjæquiuakbit Antelongiori.

! tudo, longi'tudinc 2lt;:latitudmc æquali-buSjmai'oreftjVt. 11. x. z.j*. Nabis duo funt 4, Et ter quatuor fSt iz.Hocmodonbsp;» longitudonbsp;z Latitudo

After is profunditas cætcris inteittaP lis vnitate,tarnen maiorfi fuerit, Alteranbsp;parte longiori refpondet : at plufquaranbsp;'^nitate cæteras dimenfioncs excedensnbsp;3tquipollct Antelongiori,

I

An'thmctices

AfTer 2C Latcrculus Alutudinc pu* gnanc

Cap. xxvin.

C^Vneusfeu Cuneolus numerus foli* dus eft, qui quum fecundum fuas vnita*nbsp;tes ritedifponitur, dimcnfiones omnesnbsp;habet tnaequales, ut, 14. Cuius later*nbsp;funt, 1,5,4,Nam bis tria funt. 6. Et qua*nbsp;Cerfex funt x4.Hocmodo.

Cuneus opponitur Cubo, De Paral 1 el cp 1 pedo.

Cap.XXIX.

Arallcicpjpcdus eft folidus name' rw quinumens plams qui dem 6C aequa*nbsp;hvnitatun»

Epitome.Lib. I. XXXVII. liunitatum intcruallofcpararis, fcdnecnbsp;prorfus æqualibus nccprorfusïnæqua-libusconnnctur. r e.cuiuslaterafunt,i.nbsp;J. J. Nam bis tria funt 6. Et ter fcx funtnbsp;is.Hocmodo.

longinido

Latitude)

Altititdo

Cartciiim Parallclepipedus fexmo' dis potefi: euariari,quorum primus ellnbsp;quum longitudo minorcft:æquales au-temcætcrævt i S.cuiuslatera, vtpaulonbsp;prius dicflum eiïjfunt z. 5, J.

i.

?

longitudo Latitudonbsp;profunditas

Xrithmedcca

Tertfus fit per mfnorem latftudinem Sc per longit. ac Alntud.a:quales vt. t 9nbsp;cuiiu latera fût t.^.Namcerduofuntnbsp;«sJEttcrfexfunc t s.Ita,

Dicendum er-

i profund-

Qüartuseftquum' longitudinepro' fimdïtateque aequalibns latitudo maiornbsp;«ft.vt, ( a.Cuius latera funt i. 3, ».Quianbsp;bütria

EpiwmcLib. T. XXXVm. bis tria funt ó. Et bis fex funt » i. Hocnbsp;modo»

* nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Diccndum

î profund.

Quintus eft quum profunditas minor eft æqualitate longitudim's 2lt; latitudi-tiis.vt 18 .cui'us latera faut, j. 3. x. Qui»nbsp;terniafaciunt p.Et bis nouem funt, » 9•nbsp;Sic»

?

3

ft

An’thmcticcs

Sextus eft quum acqualitatemlongf-gitudinis amp; latitudim's profimditas ex-cedttvt 1 x.cui'uslaterafunt. 1.1. j. Si-qiu’dc bis duo funt 4. Et ter quatuor funt ax.Hocmodo.

longitudo

Lahtudo

Alntudo

Proprietates.

Parallelepipedi in infinitum extru-(fti.non conueniunt.

Vnde SCa pyramide manifefte diffè' runt,

Omnis numerus folidus,Pyramide dempta, æquidiftantibus fuperficiebusnbsp;continetur.

Non tarnen omnis numerus folidus Parallelepipedus eft.

Parallelepipedus aCuneo pariterS^ Cubodiffert.

ßcxW®

Epitome Lib. I XXXIX Sextus ParaHelepipcdus modus eft,nbsp;ütalîcr.

DeCubo,

Cap,XXX.

C Vbus eft folidus numerusplanis Ct atquisfex defcriptuSjdimêfioncs omnesnbsp;«1 fe habés æquas,vt 8,funt z. x. x. Namnbsp;bis duo ftmt 4. Et bis quatuor funt 8 »nbsp;Koepado.

Diccdû igicat bis duobis^

inuentio Cub/,

Digeftisatemarioimpanbus, fi duo Prioreszpofteatrcsjdcinde quatuor,nbsp;^oiiiungantur Cubosprofcrent, vt.

Omnisnutncrus infcbis duclusCæ bum ftatuitvt Bi»duo bis, funt 8 . Tcfnbsp;!triater,funt xr.Quatcr quatuor quaternbsp;funt 64. Quinquies quincp quinquies,nbsp;funt IX y. Dehac re vide numeri pradli*nbsp;ci caput.

Proprietät es:

Cubus in cubum dudlus,Cubum pro créât.

Cubus in non cubum du (flus,non cti' bum gignit.nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Cubu»

EpîtomcLib.I. XLII, Cubus non Cubum numeral«, fc-cundum non Cubum ipfum numerat.

Si cubi commenfurabiles fuerûit SC eorundem latera.

Numerus habens fe ad cubum vt cubus ad cubum. Cubus cft.

Si numeroram ab vnitate continue proportionalium fecundus ab vnitatenbsp;foerit Quadratus,omnes erunt quadra-ti.quot ß idem fuerit Cubus:et cxteri eunbsp;bi erunt.

Si Quadratus fuerit Cubus: Latus Quadrati Cubas erit, latus ucro Cubi,nbsp;Quadratus,

Omnium duorum folidorum pro-portio vnius ad alteram eft t ûcuti Cubi ad Cubum.

Ex ducîu Cubi in altéra partelongf-Orem,nunquam produdtur Cubus.

Denumero

fcquiim ducitur, in fequocp redit.vt f. Nam quinquicsquinqj funt, x y .Ita SkT 6*nbsp;quiafcxics fcxfunt, jó.ïtavcro didlusnbsp;eft^quot in cum terminetur rcdcat nU'nbsp;metum pei qucm multiplicatuseft: in'nbsp;frar circuli cuius circumferentia in idefl»nbsp;circumducitur pun(flum.Idcm Sphscnbsp;ïicus, SCfoifitan aprius, appellatur,»nbsp;S phæra in qua fiipcrficics, quæ vna tan'nbsp;tu ni eft, in fc ipfain reuertitur.

HacRcnus denumerenimTheO' rijs nunc de coixin-dem Praxi.

Epitome Lib.IL XLI.

DE NVMERORVM. NPraxi,CapJ.

VMERORVM Praxis nihil aliudcft.quam numeii adaliquodopusnbsp;faâ’a per fuppurationem accommoda-tio.Eftq? duplex, vnaquæ fcripto. alte-raquæ fît cal culls, Illam Figuralem hacnbsp;linealem.ambas vno nomine Al gorith*nbsp;mum vocant.

Figuralis autem eff,cuiusnumcrino-tis,£C charadferibus Arithmeticis rcprx fentantur.Charadferesquibus omnis nunbsp;merus exprimitur, funt decern diftinctenbsp;^figillatimpofjti.vr, ti^4f6-789onbsp;Hæcpoftrcmafola, CCpcrfenihil quid?nbsp;lignificat alijs aût,adiun(ftafi fuerit figninbsp;heats autftius reddit. Figura nihili, cir-^ulumôdà fortaflîszyphram no-*^’nant. Hisdccem charadferibusLati-

numerant.Fîebræi vero et Graecifuas ^dexprimcndu numerû literas aeômo-^ât.fût ôdaliæ numeroi^î figuræ quas fc-^uens typus demonfti'at, G

I

\t nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Cap.II.

i \ VmertB prafticus eft triplex. Dtgi luseftomnis denario inferior, vt,

J 67 8 9, Articuius eft omnis in dec? par tcs æquas diuifibilis, ita vt pcracfta diuiflnbsp;onenihiiremaneat.vt, 1 •. io. 3 o. 100,nbsp;»1C. loco,âic.CópofitusflucMixtu»nbsp;Q i eft

Arithmctices

eft qui digno SC articule confiât. Vf, i ».

11. X ». X1. Et fane omnis numerus inter duos proximos articulos comprathcii'

feptê.NumcratiOjAdditio, Subtraeftio, MiiltiplicatiOjDiuifio,Progreflîo,ÔLrgt;nbsp;dicuminuentio.

Dcnumcrationc

Cap. un.

N Vmeratio cft cuiufuis numcripcr fuas figuras dcpitftio. Hæc docet iiuniC'nbsp;rïim propofitum fignare atep fignatiU^nbsp;riteexprimere.

Ad hanc numerorum fpecicm præcfi pue duo ncceflariafunt, ordo fcilicct^^nbsp;locus, Ordo quidê eo enim rétrogradé'nbsp;J.àdexrra finiftram verfus numerandi’nbsp;ferc in Mathematicis vtimur,tradunt

Epitome. Lib. TL XLIIL tores hums artis Arabes co modo fuas,nbsp;Vt Hebraeos fuas depi'ngereliteras, vndenbsp;gentis forfitan autoritäre fumpta, is or-do haflenus obferuatur. Locus deindenbsp;numeratione promouet, quaclibet enimnbsp;figura in primo loco( or dine retrogradonbsp;fcruato) pofitafemel hoc eftfimpliciternbsp;fefignificat,vt «.in fecundo dccies.vt,nbsp;» o.decern, 3 o.triginta,4o.quadragintanbsp;8 o .olt;?Toginta,quia oóuaginta Valla nenbsp;gat dicendum. In tertio loco centies, vt.nbsp;» o o .centu. 3 o o.trcccnta.40 o.quadi in-genta, S o o .olt;fbngenta,£Cc.In quarto denbsp;nicplocomillies,vt «ooo.mille. 3000.nbsp;tria millia,40o.quatuormillia. 8000.nbsp;odomillia.Proindchicnumei-us, 1118nbsp;fignificat mille,centum, deccm S^oclo,nbsp;tot enim annis poft natum Chriftumcznbsp;lapfis condi cacpit Fi iburgû à tertio Bc'nbsp;teftoIdoduceZaringiæ.Qiiot fi pluresnbsp;tdfint figuræ, tum quarta vt prima mil-lenaria ponitur, Qiiinta denaria millc-G 3 narra

Arithmcticcs

nariafexta, centenaria m Hl enaria, oda^ uadca'es millies milIenaria.Fitautctnnbsp;tftacc progreflïo fic vt fequentts, adim^nbsp;mediate anteuertenté ratio fit décupla,nbsp;Vnde Placcntinus fccüdum Graccos it»nbsp;difponit, ncmpcquot in pn'maregfoncnbsp;finifiram verfus numerus dicatur Mo*nbsp;nadicus,infecundaDecadicus, in terti»nbsp;hccatondadicus. Inquarta millerefidc'nbsp;at.In quinta dccies minc,fiue Myrias.Irtnbsp;fexta denac myriades.In feptima centiesnbsp;denae myriades, in odoua mille myria'nbsp;des.Innona dcniqjdenamillia Myria'nbsp;dum. Notandumetiam hoe loco vetc'nbsp;res vitra fextâ regioné, hoc eft, centen» ;nbsp;millia rarifli'meprogreffbs cfle.Xerxi*nbsp;Perfanim régis terreftrem exercitS nUquot;nbsp;mer o fuifle centum feptuaginta Myria'nbsp;des,id eft, dccies fepries cêtena millia tc'nbsp;ftatur Herodotus, Praeterea Darius, tc'nbsp;fteQ.Curtio in belluduxit, 107* »o»nbsp;hoc eft

/

Epitome Lib. TL nbsp;nbsp;nbsp;nbsp;XLTÎIT.

boe eft, decics centena milia faptuaginta Vnum milliaet ducêtos,virismulieribus,nbsp;fpadcnibus liheris cónumcrans. Infanbsp;cr snumeronimlibris legnnuscmnesfinbsp;lios Ifraêl adbcllöaptos!, SCvigintiquatuor a nnos hab nt cs tuifle numero 6 o jnbsp;S o. ApudCiceronem Aceufationumnbsp;inC, Verrem tertio legitur fequês is numerus 15:45416. Hoe eft, quindecicsnbsp;Centena qu adraginta qu in qp m il b a qu a-dringentaffCfedecim. Item 1x35416.nbsp;id eft, vicies bis centena, triginta quinc^nbsp;milia,Præterca in Macrobio legt-musita 4809000. id eft, quadragics o-dies centena millia.Et 3017000 .Hanenbsp;fummam it a verterc licet trccentics fenbsp;mcl centena feptuagintamillia.Hæcnbsp;breuiter quidem £lt; concinne dicunturnbsp;G 4 cninia

An'thmeticcs ommatbreuius autcrr. nonnulli num eranbsp;tioncm inftituifTe vidêtur vtprovidesnbsp;Seftertium.decies centena mi'llia Seftcrnbsp;tiorum.

Numerationis diffi'cultas in latinapro nunefatione fita efTe vi detur. itaeß numcnbsp;ros cautecxprimamusneautcum Albanbsp;nisinfcitæ,autcum Corœbo ftoh'dita^nbsp;tis incufent nos quibus nihil,quod fynccnbsp;rum eft,placet. Numeros itacp ad ccntc'nbsp;na millia referas,Hoc eft omnium e'x.cc'nbsp;dentium prolationem ad centenamillianbsp;difponas.vt, « o oo o o ®. fecundum craf-fam vulgilatinitate eflent mille millia,nbsp;quæ tarnen mul to latinius et terfius dixenbsp;ris,decies cetenamillia. Itaquoc^exentnbsp;plum de numerofiffimoXerxisexercPnbsp;tu paulo prius dedudlu ex interpretationbsp;ne Budei continet decies fepties cen tenanbsp;millia. Fit aute iftæc in numerorum ex-preffio commodifftmeper aduerbia.

Hoc loco opereprteciü eft ad vnguem nolle

Epitome. Lib.II. XLV fiofTe V era in prolationem Cardinaliumnbsp;Diftnbutiuoi'um nominum ordim's Renbsp;ktiuorum numeralium Multiplcatiuo-rum à relatiuis venienrium, Aduerbioijcnbsp;numcrandr, quorunda lt;icnicp in Anusnbsp;amp; Anus finientium.Inter hæc alia vent'nbsp;untintegre:aliaverofyncopata.nbsp;Hadlenus de illis quæ inreda prolatto*nbsp;He obfeniari debent.

Canon generalis exprimcH' dinumerum.

Generali's circumfertur régula qua pH-ruum huius rei penitus rudes ceu bacillo •nnixi vtantur, tn ea tarnen diuimmoranbsp;non uelim .Pn'ncipio fumant tres pria»nbsp;tes ex Alphabete literæ fcilicct, a, b, c,nbsp;C)einde fupra pn’mam figurant ponaturnbsp;Â,fupra fecundâ b,fupra tertiam c, Quarnbsp;tahabeat nirfus a,quintab, fexta,c, lt;S^c.nbsp;^0 modo vt fingulæ fi'guræ hanim literanbsp;turn unam fupra fe habeant, quo fado,nbsp;^nittca (demptio primo) millenan'um

G g reprefentaty

Aiithmctices fçpratfentat, omneb, num cru ßgntPcafnbsp;infra centum, omne c, cent um. vbi vetonbsp;a,amp;b, conuemunt, fimul expiiman.ur,nbsp;niûbjfubfeziphram habear.vt,nbsp;baebseba

4 • 8 a ) 4 Ó fcxccnties quadragies feoKl cctcnamil*nbsp;lia, odngmta duomûlù, trcc€nta,qurnbsp;drag/ntafcx.Ita,

c b d lt; ba ^85408

Scxfngeta mill la, o Aoginta tria millia, quadrïngentaô^ odo.

Ab'aregulacft, vt fupra quarts queiû que chäratfierem punÄus locetur.

De Âdditione

A nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Cap.V.

Dditioeft numeronimpropofitorl um m vnam fummam coIletQio. Hancnbsp;alij compofitïonem vocan*-.

In Additione duo numerorum orlt;l* ne# funt, primus qui amp;fuperior SC Nü'nbsp;merus, ciîifit additio nuncupatur

EpitomcLfb. IL XLVL ter fuperiori fecundum fuas figuras di-di'rec^le fubfcribitur, ÔC dicitur inferior finbsp;ue numerus addendus.Tn ordinibus autcnbsp;prima figura dici'tquac ordme rétrograde feruato, piimacft. îtaque fi duos nu-merorum limites,in vnam fummam colnbsp;ligcrc voluen's,primam figuram ordimsnbsp;hifcrions fub pn'mam fup crions diïecT'enbsp;ponas,fecundâ fub fecunda,tcrtiam fubnbsp;tcitia,amp;c. Qiio fado,lineam fub numenbsp;ro addendo ducas fub quam numerusnbsp;produóusexadditione limi’tum fenbanbsp;lur. Addatur ergo prima inferior pri-fuperioriamp; maenumerum ex Additi'o-hefadum dircdcfub lineamponas,nbsp;ï^cinde fecundam inferiorem feeundænbsp;fuperioii fimih'tcr adiungas, ôifprodu-lt;^um fub inferiorem ÔC Imeam ponas,nbsp;Eodem modo ÔC cum cætctïs agas.nbsp;Et hoc verum eftjfi ex additio nein-ferions adfuperiorê producitur nume-tus vnico charadere fcribaidus, vt.

A rühm et ices

. é 1 4 Numerus cui'fit additio ^61 Numerus addcndusnbsp;’ 9 8 6 Numerus producfïusnbsp;Siucroex Additïone proueniatnume'nbsp;rus duabus figun'sfcribendus,prima fcr^nbsp;batur, altera teneatur mente vel tabula»nbsp;Cc proximo figurx Numeri addédi iuft'nbsp;gatur.vt,

4682

9 * t vnt’taa

7 9 7 3

Figurât fupputatæ non funt delen d^^ fedtranfuerfis fignandæ virgulis, vter^nbsp;rorc fadlo numerus non abolitus rcco'nbsp;gnofcipoflît.vt,

974

Num crus addcndus pauciores interdö habet figuras quàm numerus cuifitaddlnbsp;tiü» vocatur^ iftæc additio truncatanbsp;concile

Epitome Lib.II. nbsp;nbsp;nbsp;XL VU»

Concifa. In hac fupcn'crcs figura: quibusr hulla ex infcn'oribus correfpodet iub liquot;nbsp;iieam ponantur.vt,

iff nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;.Y

Jèr X

848$^

Si infuperiore dfltaxat zipbrafuerit, fciferioris figura fub lineain ponatur.vt,nbsp;-S' o

* ° 4

Sin a«tcm inferior circulus fit fupcri-. Oris ordinis character fubfcribatur.vt,

4- o

Porro vtracp ziphrahabentesiphra fubfcribatur.vt, o

4' o 90

In fine charadcr uon fciuatur fed

Anthmeti'ceß

fcribf debet, vt,

-r Aquot;

» 1 i 5

DeExpericntîjs fiuepro-bationibus.

Eonim omniû quæ ïam de Addiïio' fie diximus, omniû item quæ de fubtra'nbsp;ôione Mulnpb'catione amp; Diuifionc diquot;nbsp;cemusjcertitudmem fiueexperientiâ trtnbsp;fcusmodisacciperepoteris. EamepeX'nbsp;pcrientiam dicimus quam alq prob atiO'’nbsp;nem vocant.Probatun'tacp Additiopefnbsp;fubtracîlioncm, per cxpcrientiam dein' ,nbsp;denouenan'am SC feptenan'am. Defubnbsp;traAione agâfequenti capite. Nunc att'nbsp;tem de cactens probationum formul is.

Complt;’'

l

EpftomcLib.IT. XLVin.

Ccmpofitio probat Nouenariæ.

Principle» fiat duarum linearum inter f^dio per modum crucis in hue moduinnbsp;Inhui'usintei-fedionis angulosnu-tneiilocandifunt. Notandum autem innbsp;proba Noueftari'a omnes figuras, quo*nbsp;loco pofitacfint, numerû digitumnbsp;teferre, lamigitur in fupenore numeronbsp;(additfo fie probatur) debent nouem,nbsp;quoties poflöt, abfj ci. Sgt;C reli/lus ( fi quisnbsp;eft) numerus in angulu crucis obtufumnbsp;dextram verfusfcribi, Simflimodopo-ftca cum addendo agendS ent : relidu»nbsp;autem (nouem adielt;ftis)in oppofitumnbsp;alteiius numeri angulum ponanir .Hocnbsp;falt;fto figurât vtriufq) anguli coniungannbsp;hi^amp;'quod prouenit in fuperioremf no-Ucm abietftis) fenbatur angulum, HuiCnbsp;dcnique relidus ex produce in inferiorem crucis angulum pofitus, arqualisnbsp;fit:Hælt;

Aritbmeticcs

*

fitjHæc omni (fiiperioris videlicet in ferions æquaîitas) fola experientia eftnbsp;amp;probatio.vt

Quot fi in probx NouenariarelicQus charac^ler fitnoucm. Circulus pro prO'nbsp;bainangulum ponitur.vt,

Compofitio Probe Se-ptcnariæ,

Quemadmodum in Nôuenan’a pro* ba character 9-ita in Scptenaria7. abij'nbsp;citur, Cedeo, vt fequitur modo, Præfcri'nbsp;bcndi funt numeri, feptem vnitatibusftnbsp;cxccdcntes,quos feptenariosuocareb'nbsp;cetita

Lpitome Lib. IL XLIX. Cet.Ita.7.»4. z i.z 8, 5 ƒ .4z.49.5 ó-öj.nbsp;7 o .77.84.91.9 S. Illis hoc ordinc dcfcrinbsp;ptisjfumenda eftproba in A dditione prfnbsp;num denumero fuperiore,duælt;p poftenbsp;riores ( ordine retrogrado feruato ) figu-tx primum fic abfoluûtur. Copulanturnbsp;in haeprobafeptenariafemper duæ pronbsp;ximæ figuræ,quarum prima digitum alnbsp;tera repræfentatarticuIum.Sumptæ venbsp;ro figuræ ad feptenarios ftatim côfcran-tur,interquos,fi inuenræ fuerint, probanbsp;nullaeritj Qtrot fi inter eofdem non nu-merentur, colligenda eft fumma vnita-tum,quæ inter figuras eft fumptaSjfit nunbsp;rncrum in ordine feptenariorum inferionbsp;rem. Diftantia deinde collecfta digitonbsp;fupraponatur prius accepto, óóadditanbsp;proximac figuræ fubfcquenti,denariumnbsp;refert.ficcp rurfum duas habebis figurasnbsp;adfeptenarios conferendas^quas etiam,nbsp;Vt priores, examinia.eocp modo ad finenbsp;'^ftj ordinis fuperioris agedum erit. Fi-H nisau-

Arith'fnct/ccs

nis autcm proba tantum in angulumpo nitur. Similipoftea modo numerus ad'nbsp;dendus examinctur, cuius proba quocfnbsp;finah's oppolîtum angul2 occuper. Hasnbsp;angulorû probas ambas lûgi'tô, ÔC quodnbsp;prouenït (abiedo feptenano)in fupC'nbsp;rioi cm ponatur, angulum, cuiproduiî^^nbsp;proba correfpondeat.vt,

3 4 « 3 t

8 7 9 S 6 6

«443^9°

Si charaifler 7.in finerelinquatur,z:*' phraponenda erït in angulum, vt.

Sï fupciior angulus Circulum ha.bi^^' n'tjcundcm quocç inferiorem haberen^quot;nbsp;cefle eil. vt.

Epitome. Lib. ÎI.

» ?

648

* nbsp;nbsp;nbsp;nbsp;9

“ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;0'

9 » 7 De Subtradioncnbsp;Cap.VLnbsp;Svbtraflio eftnumeri à numero abfa*nbsp;^io.Hancalij Subdult;ftionem nominât.

Infubtractione,vt Additione,duo nu *ïicrorûordihesfunt, Superior, quidicitnbsp;ïiumerus à quo debet Hen fubtradtio.In-feriorfupen'oridiredtefubiedtus,quivo-^atur numerus fubtrahêdus.Ex his duo-bus tertius elicitur numerus fcilicet relf-lt;?tus fub lineam, vt in additione, ponen-dus.

Notandam tarnen quotfubtrahendus ordinifuperiori, vel par vel ipfo maiornbsp;eße debet. Maior enim à minore fubtra-hipotefttninime.

H a Subtra*

Ärithmeti'ces

Subtrahere fi uelismimieronim ordi' nes,vt in Additione obfeiuatum eft.de'nbsp;bito modo ponas,ita ut figura prima in'nbsp;feriorisftctfubprirria fuperioris, fecuu'nbsp;da fubfccunda,tcrtiafubtertia, quarts»nbsp;fubquarta, ÔCë» Quibusitadifpofitisft'nbsp;neä fubij cias, fub qult; relicftu fcribas .Ita'nbsp;queprimam inferiore à fuperioreprimanbsp;fubtrahas,amp;quodremanetjfubtus lineînbsp;dircefte ponas.Dcinde fecundam à fecunnbsp;da,tertiam â tertia fub duc as ÔC relicftuinnbsp;vt priusjfubferibas, Eode modo amp; cun*nbsp;cæteris agas.vt, amp; 6 4 i

4 ? «

X z gt; 1 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;*

St inferior maior fua fuperiore fueri't» di ftantiam inférions à denario fuperiorinbsp;addas, ÔCproducftuin fub inferiorem Înbsp;nas.Etquotiescunrpdi'ftantia accipitüf inbsp;fequenti ordini vnitas addatur.vt,

0 nbsp;nbsp;nbsp;%

EpitoraeLîb, IL ’ LT. quot;Si fi gura.cui vnitas additur,foeirt chanbsp;talt;flcr p.dïftantianulla erit. Proinde fu-pehor inuan'atafubfciibat, proximecpnbsp;ftqueini, quafî diftanaafuifl'ct accepta.nbsp;Vnitas adiungatur.vt,

6^4 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;,nbsp;nbsp;nbsp;nbsp;« O O O

ƒZI ~ nbsp;nbsp;nbsp;999

9 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;.nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;¦nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;1

Sicharat^cr fubtrahendi ziphrafiiê-M't,fuperior fimpliçiter fiibtush'neâ po-îîatur. Quot fî ambæ circulares fint,zi-phra iti'dcm fupponatur. vt,

6 14 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;6quot;nbsp;O

J O 5 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;fnbsp;°nbsp;3.

__

1X1 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;I O

Siparâ pari fubtrabatur,2iphra habe atur pro relido, infinem tarnen ziphranbsp;hunquam ponitur.vt,

6 t t

411

Anthmcticcs

E ft Äf alia vulgatiftîma gaidem Cub' tralichdiratio, vbi, ft fubtradio fieri ne*nbsp;qucatjvnitas afequcntefupcriorcs ordi'nbsp;ni’smutuo fumitur.vt,

• 4 '• lt;5 •

1 I r Ó

» 9 9 i

DcProbationibus fub-tradionis.

Expen en tia Subtradionis fumitiir primo per Addinonc velut oppofitarnnbsp;fpeciera,ita, vtfi fubtrahendus additusnbsp;fuen'e relido, numerum fuperiorem iC'nbsp;diïeneceflceft.vt,

• 5 Ó 4. i

• 6 4 t

lilt

8 641

Probaturamp;^ Additio per Subtra' dionem,vtfi alter numeronim ordo»

produd®

Epitóme Lib-ÎI. LIL produdto fubducatur, alterum relinquinbsp;nccefleeft.vt, 6x4

568

7 9 4 fopen'oc

794

414_____

368^

Secundo probater Subtracflio per ex pcrientiamnoucnanäjpnmaautem pronbsp;bafumiturdefubtrabcndo. Sccundadenbsp;rcliclo; Ambepoftea lunguntur, SC connbsp;iundlum, nouenan'o abiec^ö, probac fu-pcriorisordinis correfponbebit. Proba-tio igitur cum Ad ditionc eadcm cftjniflnbsp;quotaliaordinum ratio eft^

Probatur tertio perprobam Septena riamficutpernouenariam quantum adnbsp;humerorum ordinesattinet, fuas tarnennbsp;tnteiim proba feptenaiia conditiones’nbsp;cbferuat.

Arithmetics

Excmplum»

.4- o o o Jf 4^ Hf-.7-^

Epitome Lib. II. LIU; tfts. producere. Itac^odoad4.eaproportie eft,quae eft 4.ad z.

In feducereeftmultiplicare.In multi plicatione prior numerus p ef aduerbiuninbsp;exprimitur alter uero fimpliciter.

Antequam ad generalem multipli-candi formulam veniamus, duorum di-gitorum multiplicationem, vtneceftk-tiam,traderelubet. Duorum itacp digi-torum pro pofttorum fummî feire ft uo-ïueris,vtriufcp à denario diftantiamere gionelocatam femel in fe ducas, ÔC pro-du(^urn linear ducQ^ac fubijeias. Deindenbsp;Vnius diftantia ab alterius digito tranf-Uerfim fubtrahes, quodquerelinquitur,nbsp;produdlodiftantiarumpoftponas fî^apnbsp;parebit digitorum fumma, vt.

QuofiprodudS cxmuftipïicationè H f ° diftark

Arithmetices

diftantianim duabus fcribcndum fitfi-guris. Primafcribatur, ÔC alterarclidó tranfuerfa: fubtracïlionis numero adda-tur.vt, .

Alius modus Multipïicandi digitos.

Si duo proponantur digiti,quorû furrt ma fitinquirenda Minoris acci'piatufnbsp;articulus. Deinde difFerentiamaïoris di'nbsp;giti â denan'o in minore ducatur digitS,nbsp;quodqj proucnit à minoris articulo fub'nbsp;trahat Sdremanebit digitorâ fûma.vt.

»4

Modus alius.

Propofîtorum duorum digitonJrt* ùiæquaJiüm fummam hoc modo inquiquot;nbsp;.nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;respond

Epitome Lit.!!. LTÎTT. feSjponeminoris articulum a quo miho-fem digitum toties fubducitp quot vni tanbsp;tibus mai'or digitus, à i o.abcfr, ÔCin refinbsp;duohabebi«! fummam.Idem quoep fit innbsp;digitis æqualibus, altero in articula for-mato.vc,

pi So

7»

NcnnuHi digitorum dudlum ex men fa, vt vocaturPythagoricapctunt cuiusnbsp;hæcformaeft.

Äritbmetices

In multiplicatione rétrogradas obfcf üaturordo.

Vnitas nccmultiplicatnec dniidit.

In omni mulnphcatione prior nuiuC «US per aduerbium expnmitur.

Canon Generalis.

In multiplicatione duo quocp numc' rorum ordines funt.Supen'or qui multi'nbsp;plicandus,infcn'or fupen'ori direde fubnbsp;ie^us,qui MuIriplicSis nominatur. Subnbsp;ordinibusi'tac^ ducaturlinca, fubquaiunbsp;tertîus ex numeroiö du»fluinucntus fcrinbsp;batur. Deinde primam inferiorem dult;^nbsp;inomnes fuperiorcs oïdine retrogiadonbsp;feniato, SC produk7û ponito dirc(?lefubnbsp;lineam, poftea fecundâ inferiore eoderiinbsp;ordine SCmodo in omnes fuperiores du-'nbsp;casproducffû ealegefub lincSponas vtnbsp;locus produffli loco charadlcris multipl*nbsp;câtis refpôdcat.Similiter et in alîjs agertnbsp;dumeiit, figuræ deinde ex numerorumnbsp;dudü

EpîtomcLtbJI. LV ^ndufub lineam pofitæ per additioncnbsp;collïgendæ func m vnam fiimmam.

8 4 óMultïplkandus _____r 4Multiplicans 3 ? S 4

*69%

a ® 3 o 4 Summa

Qinim char a der m circularem dttcf-tur, vnitascp mente tenetur, hæc eadcirt *nitasfcribi debet.vt,

4 o « Superior

6 a Inferior

3 I 6

1408____

a J a 9 6 Summa Circulus autem in circulum uelcba*nbsp;^^dercm du dus fc producit.vC,

8 o

I o

o o

8 o

$00

V . r

V . i

.,-j- Arithmctkes

Chara(flcr in Circulum duduSjCi'rcU' ïum procréât, vt,

8 o

1 ó o

Hoe loco perpcram agere videntur qüiDuplationem etiam fingularcm nu^nbsp;men’ pradicîfpeci’em ponunt.Eft cninvnbsp;non numeji pfadici fed multiplicationnbsp;nis fpecies. lam fi Duplatio feorfim fpc'nbsp;des cenfenda eft,quidj quæfo obftet quonbsp;minus triplatio, quadruplatio décupla*nbsp;tioôCaliat quæ mnumcræfunt,codeinnbsp;mine rede dici poflente*

Ducere Articulum in Articulum.

N cgi e (ff is vtriufcp numeriziphris, duc figuram vnius fignificatiuä in figni*nbsp;ficatiuain alteriusö^ produdonumeronbsp;vtritifcp articuli ziphra anteponitofuonbsp;dextram verfus or dine, amp;fummam ha*

Èpitome Lib. IL LVL Muln'plicationis modus elegans.nbsp;Propofitis duobus numens multipli-tandis pinge figuram rccrilincam quamnbsp;paruis diftjnguico redtagulis.Huius au-tcm figuræ longitudo tot habent qua-drangula quot elementa in mul tiplicannbsp;do fuerint,latitude vero tot quadrangu-lasfuperfidcsteneat quot in multiplicanbsp;lefuerint elementa. Deinde quodlibetnbsp;quadrangulû diagonali intetfeccs ex æ-quolinæoIa.Quibus ita pcracris, multinbsp;plicandum adfummâ longitudinê, mulnbsp;tiplicantem uero ad dextrSfigura: latusnbsp;exordinead quadrangulaponito itavfcnbsp;fingula cuiuslibet elementa adfua ordi-nata fint quadrâgula, tum enim primusnbsp;(ordinc retrograde notâto) tharadernbsp;mul tiplicandi idem tenebuht quadran-gulum : reliquis deoiTum fuo diftribu-tis or dine, Multiplicentur poftea fin-guli charaderes, per fingulas multipli-cantis figuras, â^produdi numcri proprijs

Aritbmctices

prijs infcribantur quadrâguh's, fta, vt di gitifub diagonali ei'ufdem quadrangu-lijarticuli uerofupra diagonalem locennbsp;tur. Deinde in vnam fummam col 1 igantnbsp;finguli charatQeres diagonalibus trani-uerfaliter feiun(fli:initium autem collc'nbsp;dionis feu additionis fiat in d ex tri latc^nbsp;ris ima parte. Producfïu colledionis pO' ,nbsp;natur fub diagonales, vt.

Epitome. Lib.TI. nbsp;nbsp;nbsp;nbsp;LVIL

De Probationibus Mill tiplicationis.

Expericntia Multiplicationis. vtall ftnimjtnplex eft. Sutnnur cnim aDiui-fionc, Noucnan'a SCfcptenana. Déprima fequenti capftevidebimus.

In probanouenaria ita agendum. Pri ma fumitur de MultiphcandOjSecundanbsp;de MulnplicâtCjquæ in angulis in ft duel æ, reiedo noucnario producSt nume-rumin fuperiore angulolocandum,cuinbsp;proba Summae par eiïe debebit.

Proba Scptcnaria, vt nouenaria eft, hifi quod fuis conditionibus,vtit«r ilia.

864

2 0

f t 8 4 1728nbsp;2 2 4 6 4

I Expert

An'thmcticcS de haefequenz z464( »nbsp;ncapite. SÓ4nbsp;Prima Dim'fionênbsp;xz464(8^4nbsp;Expenbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;O ZÓ

rien- Secun. per 9 °x\° tia..

Tcitia 7 y/\?

1

N otandum quod in proba nouenari^ feptenariafivel Multiplicandus vc^nbsp;jMulripIicans ziphram inangulum pO^nbsp;fucrit acutomm angulorû probac itiddî¹nbsp;circul ares crunt. vt,

Exemplum de Multiplicante. 4^4

? nbsp;nbsp;9 O

î y 9 Z nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;O

* 7 'S 4 ¦

¹

9133

Exeit»'

EpitomeLib.n. LVin. Excmplum de Muïtiplfcando.

DeDiuifione»

Cap,vnr.

Tuifio cft ex duobus niimens prc-Vofitis j'nuentio cuiufdam terti)\quivno propofitOjtoties cffe depræhendit,quotnbsp;’naltero vnitates funt. Eftque Diuifionbsp;Multiplication! plane contraria, namnbsp;Siuod hxc difpergit illa colligit.

In Diuifioneduonumerorïjordines ^unt, Superior et inferior,ille diuidêdus,nbsp;^ic Diuifor feu Diuidens appell at, T er-tius per Diuifionê inuentusaduerbij nomine vulgo Qiioties à placentinoDiui-foriusnominatur.

J Arithmctfces

In Dmifione non 1 inea fed femicïrci^' luspoftnumcrorum ordines dextravefnbsp;fuspHigifoictj'nquê Qiioties fcnbitnr»nbsp;Ad Intclh'gendam Diujdendi ratio'nbsp;nemfubrciiptænotentur

, ¦ Hypochefes.

Tn diuifione incipiendum eft finiftr® lafcic.

Vln'ma Diuifoi is ponenda eft fub vî jimaDiuidcndi. Et hoc quidem verun*nbsp;fi vlfima Diuifonsnonfuerit maiorvîUnbsp;ma fibi fuprapofita. Nam ft maior exti'nbsp;tent,fub pcnuln'ma di'uidcndi locetur vlnbsp;timaDiuifon.vt, 4 6 x

;

4 O X ƒ

- Non debetmaior nouenan'o fcmel ift (cmîcirculum poni.

• Diuifor poft quami ibet operationcrrt debet ciTe maior numero ftbi ftiprapofnnbsp;'nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;tore'

Epitome Libi IT. LTX torefpedufui. In fine ante rcCpccruto-tius Diuidcndi.

Poft vnam opcrationcm, varietur Di Infor per vnâftgurâjid eft,in fubfequen*nbsp;tem locum ponatur.

Si in media operatione aut fine Quo ties inueniri nequea t,ponatur zipfira adnbsp;quotientem priore. Et varieturDiuifornbsp;per vnam figuram dextram verfus.

SiprimaDiuiforis fub prima Diuidc di conftitcritDiuifio pcrada eft.

His omnibus notatis poneDiuiforent iuxta fecudam hypothcfin fubDiuiden-dum, Sé vide quories vl tima Diuiforis innbsp;numéro fibi ftiprapofito habciipolTit,!-beat,toties in fuis fuprafci iptis inueniannbsp;turfiguris »Qiio perfpedo,poneQito-tientem in Scmicirculum quem deindenbsp;per totum Diuiforcm mul tiplica, fiCpronbsp;ducftûàfîguris Diuifori fuprapofitisexnbsp;Ordine fubtrahito. Relicftu vero, fi que

Anthmctfc« ¦*

habescx fubtracSione numcruiadicfii® Diui'dendi figuris fupraponito.Hac pri'nbsp;mafcihcet operatione peraó:a, vanctütnbsp;diuiforper vnam figuramjhoG eft,Diui'nbsp;fons prima figurafnb fuperiorem fefG'nbsp;quentem ponatur. fecüdainferiorisfubnbsp;fupen'ore fe fequéte, amp;c. ita tarne, qtiodnbsp;totus Diuiforfi plurescharadcres habenbsp;at àfuo loco ponaturin proximum. Po'nbsp;fito xta^ nirfïim Diuifore alius quæra'nbsp;tur Quotiens in figun's Diuïfori fupra^nbsp;pofitis,amp;ïn relilt;fî-is,fi quæ funt,poftfubnbsp;tra(5fionc,pauloprius facflâ.Cum hocfe^nbsp;cudo Quotiêtc ÔC omnib.altjs non aliternbsp;cû primis ages. Fine vcro Diuifionisnbsp;iuxta fcptimâ Hypothefin cognol^-vt, .1-nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;t-

(47 X nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;O

-V

Porro fi numerus in fine relinquat, irt quo Diuifor haben' non poffît,vocat refi

EpitqmeLib.IL LX «ïuS.Idc^fcribendS eft poft Quotient?,nbsp;dexträ verfusin fuperioreloco cS Imeanbsp;fubiaceat, fub qua ponatur Diuifor, quinbsp;Vniusintegritotpartes,quot vnitates hanbsp;tet ftgnificat. Refiduiïvero femper totnbsp;Piuiforis, ld eft, integri partes numeratnbsp;quot vnitates habet.Eftlt;çrefidu2cuninbsp;tiiuifore fibi fuppofito plane nihil aliudnbsp;quàm fraA'o feu JMinutia,

Refiduû femper Diuifore minus ef-fe debet.

Refidui præterea denomihario non fit â denominatore Diuidentis,fed Quonbsp;^icntis.Idem enim SCvnius vtriufcp denominator eft.

1 ExempladcRefiduo

Refid.

* nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Hoceftduodecimvicæfi-

mæ quarte vn^usinccgrƒ.-•*¦'8-.ó-

Arithmetic«

4/«

A- ,1 ’-fi

Jir jSr .V C194 Ï7 hoc eft duodccirtt at 4-nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;vicefimæquints^

at tf- tf- o nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;vniusintegri

at af

»

Exempta fextæ Hypothefis.

t

Jtr 0 o'^ (sooi ^vnaoeftaua

, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;-S'nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;vniusintegB

jS-

^800^ nbsp;nbsp;800.0^

6 6 6 6

Canon,

Omnis numerus per atiquemmulti-, plicatusjh Diuifioncnihil habctrefiduinbsp;'fii enim produdum ex Multiplicatio*nbsp;ne per M ilnplicandum diiufumfucritrnbsp;(uhil remancbïc,vt.

De Probationibus Diuïfionis.

p. • . Y- ’ ‘ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;’ Muitipiicati®

*-t nuius fpeciei experv «, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;.

n. - 1 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;iNOucnana

tutiaeft triplex nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;o

bcptenaria

Probatur primS Diuifio per JMultipIi plicationê vr Só Muln'plicatio per Diutnbsp;ftoné. Per Multiplicationê vt fi quoticanbsp;tem perDiuiforê mulripliccs,inprodu-fto cum Addirianereßdui (li quodfue-rit^numciûhabcbîsdiuidêdûjMultiplinbsp;catioms aSt certitude cx Diuifione eft»nbsp;Nam fumma per Alulriplicantem diut-’nbsp;fa, Multipb'candum in quotientepiodixnbsp;Cit, Aut cadem per Multiplicanduranbsp;diuifa multipljcantem pro Qiioticntenbsp;ponit»vt»

I f

Arithmctices

4*^ 5 o i

604 ^34»

»4-0

»

'4- G*

*

Innoucnariapnma probafumitur dc Diuifor c. SecSda de Quotiente.Hae du'nbsp;cantur in fe, Sgt;C numerus produdi, nouc'nbsp;narioabic(S;ö,relidus cum additionerc'nbsp;fidui, fiquodhabeatur, correfpondebitnbsp;probæ deDündendo fumenda?.

Septenarïa fuasleges obferuat,aliâs » Vtnouenariafît. Notandû tame quodÛnbsp;refiduS vel feptenariuexcedat vel plurxnbsp;bus figun's quam vna fcnptSfit:fumeda!nbsp;erit proba, vt in cæteris, de eodem, quæ,nbsp;abiedo rurfum feptenano, lügat probxnbsp;Dtuifoiis

Epitome Lib.IL LXII Diuiforis as Quoticntis, Saturn dem«mnbsp;par exit probe Diuidendi»

st i-

st

* 4

Multip.vt,

X 1 S

DePro'

Arithincticcs

De progrefîî'onc,

P nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Cap. IX.

R.OGR.ËSSIO eftnumerorufli sequalitcidiftantium in ynam lummaiMnbsp;colledio.

Continua

/Arithmctica

I nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Interdfa

V

Gcorrtetn'ca. HæcTuas fpe-?es in infinitum extendit.

ProgrclTio Arithmctica continuafiuc naturabs eft vbi poft pi imum characfte-*nbsp;rem nullus intcrmirtit.v^t, i x 34.VCI 54;nbsp;é.vel ƒ 67S’9.vc1 Ó r 8 p t a j r 1 z,6(^c .

ProgrelTio Arithmctica difeontinua fiuc intcrcifa cftfiguris æqualiter inter'nbsp;ceptis numerorum ordo.vt, 15 jnbsp;gt;40 3 1 o.ô^c.

Epitowe.Lib. IT. LXIII.

De Progrefïïone Arith'

. mcn'caduærcgulæ. . lt;

» Si numcroixim fecundumPrcgi'cHio ncm Antbmeticam dcfcriptonim fcri'esnbsp;eft par,addatur primus vltimo amp;'prodtfnbsp;ôum ducatur m iticdictatcm numcn lo'nbsp;Corum : quodq? I'ndc prouenit, numero-rum difpofitorûfummacft.

i- Srvero numerorum difpofitorum ties eft impar. primus vt antea luiigaturnbsp;vln'mo, S)C per produtfti medietatcm torus locorum numerus multiphcetur, Si.nbsp;in multiph'cationis produdo quæfituninbsp;apparebic.

Numerus ferici fine locorum eft qui indicat,quot in ordi'ne difpofi to numertnbsp;Tint ,vtmhoc ordine 23456789 10 11nbsp;’ funtvndecim numeri fiuenumerorum intcrftitia.

Nunquam prætcreafïf,vt numerus Tocorum et numerus ex additionc pn'minbsp;vltfmS produeftus fimul fint imp arcsnbsp;Ambo tarnen fepænumcrb pares fuut.

Arithmetice» Excmpla.nbsp;t i j 4 ! 6 7 8.

AhæRcgulæ deProgref-fione Arithmetica.

Continua Progrcflîo in parem fi de* finit,medietatem paris ducas innumertfnbsp;qui parem immetÛctate fequitur. vt.

• i ) 4 i

I O

Continua Progrefïïo in imparem (ï definitiMaiorem imparis portionem innbsp;. totumimparem ducas.vt, t »

* ? 4 nbsp;nbsp;4 ƒ ____

InterruptaProgreflionenumeri finiquot; ente, Medietatê eiufdem paris duc in nUnbsp;merufuperiorê proximû raedietati. vt.

Epitome Lib.IL

246

w' nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;*

z 1

InterruptaProgreffïone dcfincntc ift imparem.Maiorern imparis portionemnbsp;duc femel in feipfani.vt,

I;

9 Canon De progreiïïonenbsp;Geometrica

Progreffio Geometrica eft difpofitio numeroium aliquaproportionefi exce-dentium vt Dupla T ripla Quadrupla,nbsp;SCc.

Omnis progrefli'onis Geometricæ fummafacile cognofcitur fi vltrmus pernbsp;numerum denominationis proportie-nis multiplicetur, à produ(5ro pofteaprïnbsp;tnus auferatur, SCrclidus per numerumnbsp;vnitate minorem numero proporno-nis diuidatur, In Quotientc enimnbsp;fummam

An'thmcticcs

fummam depræhcndcs

Exempta.

De Probationibos

Progreflïönis,

Progrefïïonis ceititudo tribus modi's deprxhêdi'tur, Subtradtione noueiiarianbsp;CCfeptcnaria.

Probaturper fubtraclionem. Nam ß fïngulos dan' exempli numeroamp;à fum^nbsp;ma fubduxeris, nihilquc remâferit bciiCnbsp;progrcHvijs cs.

uî4n riouenana ôf feptenaria duæ tan ' tum

türm accipiunturprobae.

Innoucnariaaccipcprobam priorein de Omnibus exempli numeris, 6^quäli-bet figura figillatim examinata remouenbsp;9.quotiespotes. Huicfi fummæprobanbsp;parfueritjbeneadlum eft.

In Septenariaita agas. Priorcm pro-bam fumito dequplibet exempli numero fiue vna figura fiue pluribus fcripto. Quas probas omnes ad fe addas, 06 in fe-ptem, quoties licet, remoue. Cüi probanbsp;fummac correfpondeat.

ExemplumProgr.Arith.Continuae«.

• nbsp;nbsp;nbsp;nbsp;nbsp;4 f 6

Prima Subtradxoncm

K Excm-

' Anïhmetices

ExcmplumProgr. AMthm.intcfdfaCi

» 46 8 to Î3 0

Prima Subtradioncm

Tertia 7

Exemplvim Progr.Gcometricar, t l 4816^1 64 J « *znbsp;Prima SubtraÆ'onem

‘Experj Secun. per 9. entia»

'Terria * 7

De Radicum Inuentione.

HCap.X.

Aecex toto numeropropofi'to

Ottadratuï”

Epttome nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;LXVr, •

Qüadratum C£tbicum,veï tadicê hoe «ft, latus maximi Quadrat! Cubkinbsp;fub propoßto numero content! ponit.

Ad cxadfiorcm huiiis capitis intelle-^um repetenda funt,quæ de numero fu perficiali QC folido fupra diximus.Namnbsp;hi fob radicem habent,

Itaqueprimo vidcndumquïdnumc-tns quadratus, qu id quadrat! radix,quid dcinde,radicem fit inuenirc. De Cubicinbsp;tationepoft inuentam quadratam radi-Cemaçcmus.

Quid numerus fît Quadratus,require ex num eris contcmplatiuis.

Quadrat! radix eft numerus qui femel infe ducit, vt 4. infe femel duco amp; pro-ueniunt i 6.huiusprodudi 4.eft Radixnbsp;hoe eft latus.

Radias igitur quadrat! inuentio nihil aliud eft,quam ex propofito numero lateris quadrat! inquifitio.

Porro Superficialis numerus eft qui K » fit ex

Arithmctices fit quadratus, Si veto femel in aïinm,fitnbsp;fuperfiaahs qui dem, fed non Quadra^nbsp;tus. Solidus item numerus eft qui fit elt;nbsp;dutftu numeri innumerû.Dudus auteff»nbsp;numeri bis fit, aut emm bis infe, fitepfo'nbsp;lidus ôif Cubicus. Aut toties tn alium,nbsp;fit folidus quide, fed non Cubicus. HxCnbsp;ex infcquenti typo clarafunt.

Semcl Se g^fitfu quadrat? Nume aut in aliumP^’^ß^’^^noquad«nbsp;tus in

nume nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;onbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;i ¦

rûdm Bis nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;fitc^fo- cubicus

citur. aut in alium^i^usS^^^çyl^jc^

Ex iam didis patet quod idem num^ tus eft radix Quadrati èc Cubici, non tanbsp;men radicis illius idem Quadratus eft»nbsp;amp; Cubicus. Huius ratio eft, fiquidem o'nbsp;mnis numerus poteft efle radix Quadranbsp;t! pariter Sé Cubici, attamen non omiri^nbsp;numerus quadratus eft aut cubicus. Ita'

Epitome LXVIL ^ue radicem quadratä clicere uel eft pronbsp;tgt;ofitinumeri(f)totus quadratus fit) latus inuenirc. vel, fi totus quadratus nonnbsp;fitjlatus maximi quadrati, qui fub totonbsp;propofito eft,extrahcre.

Ad Quadratæ radicis inuentio-nem hæ notentur Hy-pothefes.

Radicum inuentio eft quaedam fpe-^iesDiuifionis.

Vnde femidrculus, ficut in Diuifione, poftpropofitum numerS dextram versus ponendus eft,in quem radix inuentanbsp;feribi debet.

In Radicum inuentioncvnicus dun-ï^xat numerorum eft ordo.

Pracfcriptus numerus,cuius radix qua ^rata qureritur,in Iocis imparibusfignenbsp;^ur puneft is. obferuatur autem rccrogra-ordo in numerandis locis.

Quod pu(fta propofitus numerus ha-“Uerit, tot SC figuras feu digitos m fnni-K drculum

, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Anthmctices

tirculam poni necefleeft.

Slib vlumo (adfîniftrâfcilicct) piin Ào primus quæràtur digitus.

Dudus digiti fn feïp fum femper füb aliquo punó'O fiat.

Semper totum, quod eft in fe femicif culo duplandum eft.

Si àfuperiore(qui notacircdïaris) V' nitas no poteftabijcitfumatur proximonbsp;(equens,aquavnitas dempta refoluatufnbsp;' in I o .e quibus nouem in locum circula'nbsp;ris figurae fubftituas.

Si in medio aut fine digitus inueniri nc quit,ponat ziphra in femilunula, FigU'nbsp;rac aut in fine relidæ dénotât refîduS..

Si omnibus perac'lis in fine nihil rcma, netjtotus numerus propofituseft qua'nbsp;dratusj 6^ ergo numerus in femicirculonbsp;contentus eftradix fiuelatus propofirinbsp;numeri,fivero aliquidin fine relinqufinbsp;tur,totins propofitus numerus quadra'nbsp;tus non eft, QCproinde numerus femicir

Epitome, LXvni. cull non eft radix totius propofiti fednbsp;radix ÄS latus eft itiaximi quadrati fubnbsp;propofito corttenti.

Maximus quadratus radice infeipf^ ducîa producitur.Omnis enim nume-rus femel in feipfum dudus quadratumnbsp;conftituet,

Formaradicis Qiiadratac inueniendæ.

Numcrum,cuius quadratam radicemv quærisjinlocis imparibus fignatopun-lt;flisita4 16 8 Deindefub vltimopun^nbsp;^o , quaere digitum, qui femel infedu-dus del eat per fubtraêlionc vel totumnbsp;quodfub punefto SC ante pundumfini-ftram uerfus eft, vel quatum de toto iamnbsp;diSto polTis, Digi turn ergo inuentû po'nbsp;tie in fcmicirculum, eumquc femel in fcnbsp;ducas, Sc producrum refpeeftu punCbj vtnbsp;duflmneft-^ubrahas ita.

Anthmetices

JÄ-

a (à

JY

Poftea digftum in femfcircuïo dupïa, SC dupla fub proxima verfus dextra po*nbsp;neproximâ vero didmusquæ dextramnbsp;verfus fequitur pSc^mfub quo digitusnbsp;inuentus.Porro duplatum itaponatur,nbsp;vt prima eius figura ftetfub proximanbsp;poft puncftum dextra verfus, cætere ve*nbsp;ro duplatifiguræ, fi qaæfunt,ilocenturnbsp;ex ordincfub alias figuras finiftrâ verfusnbsp;Vt.nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;6

4 16 8 Clt;i

t t

Quo fado, fub proximo dextra ver-fuspundo alium quaere digitum, qui ad priorem in femicirculum pofitus àpri'nbsp;tno dudus in duplatu totum,vel drcicetnbsp;totn del cat fuprapofitS refpedu dupla^

Epitome LXIX, ti. Deinde ductus ide digitus in fei'pfuntnbsp;lotu vel propetotu deleat fuprapoütumnbsp;rcfpeôu pundi fub quo inuentus eft.

Delete refpecftu duplati eft per fubtra dlione tollcrefiguras qu^ non tantu fu-pra duplatu in ordine propofito funt ponbsp;fitæjfed etiam eas finiftrâ verfus antece-dunt. Attamen fubtradione facftaple-mnep aliquid relinquitur.

Delete vero refpedlu puntftf eft per fubtra(ftione tollere non tantu figuram,nbsp;fub qua digitus inuetus eft,fed ÔC omnesnbsp;finiftram verfus præccdcntes,Relinquinbsp;tur autem SChic fepænumero nonnihilnbsp;poft dcletionem,id eftjfubtradionê.Vt.

^43

* *

S’ (6;

jif at 4-

o

lam ergo operatio omni'no fa da eft, Radixqß muenta. Supereft vt luxta vn*nbsp;K ƒ decimam

Arithmeticcs

decimam Hypothefîn côcïudatur,abfal uaturcp, deinde excmplum propofîtumnbsp;iuxta duodecimam Hypothefin.

CaeterS fi m exemplo plura ß nt quant duopunda, cumduöbusjVtiamdiduntnbsp;cftjagito. Cum tertio autem punâ-o itanbsp;operate.Principio totum fcmicirculi nunbsp;tnerû iuxta tcnorem ocfïauæ Hypothe*nbsp;fis duplatduplatû fub proximâ vt phus,nbsp;ponas poft duplati polîtionê abus dein'nbsp;de fub tertio dextrâ verfus punâ-o quac'nbsp;ratur digitus cum quo vt prioribus agas,.nbsp;Exemplo atitêadcolophona pcrdudo,nbsp;refiduum fi quodfuerit, dextram verfusnbsp;poftQuotientem ponas, cui etiam maxinbsp;mum quadratum fubijcias iuxta » x. Hÿnbsp;pothefin.

Excmplum de tertio pundo,

a

Epttomc

* ; .

itr nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;X48 refld.

* • • ¦ • .. ”.. ¦.

. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;.é-(ii9

5x44» ma.Qua.

jêf

Quod fi datam excmplum quatuor puncfta habeatjficagito. T hbus punlt;5lisnbsp;abfolutistotum femrcirculi'numerûdu-pla,duplatû fub proximam, vt priüs locate, deinde fub quarto puneffo quære dtnbsp;gitS, quiphmo in duplatû dutfî-us deleatnbsp;fuprapofitum, f efpecn:u duplati, pofteanbsp;ducHus in feipfum deleat fuprapofituînnbsp;relpedu pûdi. Et fie de numero pluresnbsp;pundros habentc, agedum eft vt fcilicetnbsp;primo totus Quotientis numerus duplenbsp;tur.poft duplati vero pofîtionem abusnbsp;quaeratur digitus, ôôe.

Exempl uni quatuor puncîorum.

Anthmeticcs

« $ »147rcfid'

7 » 4P 4- -8*

• • nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;•nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;•

J6f nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Ctiyt

.1- :3f o 4- 46ióSoi o Cmax.Qiiadr,nbsp;•*- lt;gt;f

Dc decima Hypothcfi. Ëxemplam medij.

18^0. refiduttm

iÇi;78;i(x^o»»

ótffoot max.Qaa. Excmplum finis.

loorefid.

864900 ttiax.Quadr» Exemplum mcdij 8Cfinis,nbsp;zó refid.

400Z0

Epitome LX XI.

De Probatlonibus Iniicntio. f Ill's Qiiadratæ radicis.

T res experientias habet, Multiplie» tionem. nouenariam amp;feptenana. Pernbsp;Multiplicationemita. Ducradicem in-uentam in fe quadrate ô^refiduum addenbsp;produdo, huius dudus 6Ó propofitu ha-bebis numeriï.Si' veto nullum refiduumnbsp;fuen't.radix infedudaproducet nume-rum datum.

In nouenan'a feptcnaria folum duat probæ accipiantur.

Tn nouenan'a priorem probam accipc deRadicein femilunula quam I'nfequanbsp;drate, hoc eft, femel ducas, probæ radi-eisadde probam derefiduo. fi quodfuc-htjfumptä. Cui probapropofiti numerinbsp;correfpondebit.

Per feptenariam vt per nouenan'am probatur. Hæc tarnen fuis vtitur condi-tionibus.

Excra-

-- nbsp;nbsp;nbsp;nbsp;quot; Arithmctkes

f ' « nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;»

S 6 f o o o C930, «oorefid.

Prima Muln'plicationem

Terria 7 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;3

Vfus Quadratæ radio's eft in dimctH cndaduonim löcomm diftantia. Namnbsp;fi duoproponantur loca longitudineó^nbsp;Ïatitudine diftantia* DifFerentialongitiinbsp;dinüamp; latitudinum ducâtür infeipfamnbsp;proucnict^ quadratus numerus. Hi dc'nbsp;j'ndeQuadrati coniungantur produ'nbsp;dï-f radix quadrata quæratur. Radix in'nbsp;ucnta.amp;per i y.multiplicata,milasinnbsp;produdlo dabit.vt,

Quadrat! fundi faciunt. *79«

Radix ( 16. 15. refid,

Radix per 15.multiplfcatafacfti4® wiilas.

De Cubfcæ radicfs In-uentfone

Di'duin eft ex dudu numerf in febis vcl femel in fuum quadratum côftitui fonbsp;Iidum parftcr 2lt; CubicS, Solus enim fo'nbsp;Ildus, ÔC fi non omnis, cubfcaradfcem ha

Princfpfovfdendfjquidnumerus Cu bicus,quidCubicaradfx quid item fit ranbsp;dicem Cubicam inuenire.

Numerus ergo Cubfeus eft, qui ffc ®x dudu numerf in fe bis aut femel innbsp;fiium quadratum.

Radix numeri Cubici dicitur numc-^fta duplid dudufadus. Vndepatet

A rit hm et ices qttod numerus Cubicus Quadratus C“nbsp;andeni,vtfuprx ditîlû eft,radicê habent.

Radicem inuenire Cubicâ eft exnU' mero propofito latus elicere Cubicû vdnbsp;propofiti vel niaximi Cubi’d fub proponbsp;fitoCentern'. Nainfi poft operationemnbsp;fatftam nullum fuperetit refiduum,totusnbsp;propofitus eft Cubicus.Contra liquidnbsp;in fine remanferit,propofitus folidus quinbsp;dem eft non Cubicus.

Adillius quoep radicis inuentió-nem quædampropofitio-nesnotentur.

Numerus, cuius Cubica radix quærf' tûr,fignetur pundis in primo fcilicet loquot;nbsp;co,ô(^ fîngults millenarijs.vt,

• ? •

4- 6 X 86x4.

Semicirculus ad datu ponaturnum«' mm,in quemtot figurælocentur, quotnbsp;punda datus numerus habuerit.

Sub vltimo pundo initium operatic nis efîe debet.

Epitome LXXlIf.

Sicirti'n inucntionc quadrata tonim ijuodponitur in femicirculo, dttplâdumnbsp;ècduplatö f ub fecundam dexteram vernbsp;fus ponendum; Jta in cubica totS Quotinbsp;entis fiuc femicirculitriplandum,0Ctri-platum fub fequenti tertiaponendu eft.

Triplex in hac inuentionis fpecie,fit mulripbcan'o.Prima eft totius Quoncnnbsp;tisintotumTripIatum,m cuius produ*nbsp;0-uin .fccunda eft folius digiti vltimo mnbsp;Ucnn'.Tcrtiaeftci'ufdem digit! m feevr-bice.

Si in medio digitus inucnirinequeat, ponaturZiphrain femicirculum. Et dinbsp;miflisficutinpratcedenti fpecie, omni-bus,pergeadproximum pun (ft um, fubnbsp;quo alium digitum inuenias.Priustamênbsp;tótum,quod eft in femicirculo,tripleturnbsp;Hocautem infinefi contingat,ponaturnbsp;Vt antea,circulus adpriorem (^otien-tem relicîa: figui ac habeantur pro reftnbsp;duo.

L Forma

ÄritbmetkcS

. Fontia Cttbicæ radictst inuemcildæ.nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;f

Niimerum iuxta primampropon' noncmpimciisfignato. Subvitimodc'nbsp;mdepun(?îo quaere digitum qui dueftusnbsp;infcCubicc torumfuprapofitum.vcl dcnbsp;toto,quantü poffit del cat refpedu pu uquot;nbsp;difub quo inucrus eft.Digitu poftcanbsp;inuentum fepoiiein Semicnculu : eüti'nbsp;demcptnpla, öi^tn'plati produdumfubnbsp;proximadextcram verfus tertia poniti?nbsp;itavt prima triplat! (fi multas habeat)nbsp;figurafub tcrtiaiam didalocctur, caete'nbsp;'tac Veto præcedentibus finiftram verfü^nbsp;Quo fado, alium fub proximo dextisinbsp;verfuspundo poßtodudus intotûtrï'nbsp;platû, dcindcfolus dudus in produdi^nbsp;totum vcl quantum poffit, aufeiat refp*^nbsp;-du triplati-.poftea dudus idemdigitusnbsp;folus in fe cubice tollat fuprapofitum rC'nbsp;fpedu pudi.Qyibus tranfadis, totö f^^nbsp;midrculi numerCi tripla, ö^tiiplatjpu?'nbsp;duduiï^

Epitome' LXXIIII. dudujn numenim fub certia vtpnus,nbsp;guradexcramverfusponas. Exïnderurnbsp;fus quatre digitum fub fequentïpuncïlo,nbsp;.ô^c.Cæteiafecundum datas propofitionbsp;Jiespro tua induftria P erages.

Exemplum in quo nullum eftrefiduum.

A- Ar nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;•

Excmpluin fdidui.

’ nbsp;nbsp;nbsp;nbsp;• *nbsp;nbsp;nbsp;nbsp;. X 8 6 refid.

4op6* (max.Cub.

' L i Ex^

Arithmetic«’^

Excmplum m edij amp; finis* • • „ fnbsp;nbsp;nbsp;nbsp;nbsp;1811 refid.

tooiSii (joo

'' nbsp;nbsp;nbsp;nbsp;I ooo ooo max.cub.

Supra dicHum eft turdem numeruni cficiadicem Quadiati amp; Cubicicum vCnbsp;icQuadiarum St Cubicum ncnelTecU'nbsp;ius tale fit cxtmplum*

lt;4905. gt;790 rcnd.(Qiad, ^46^1509 max.

• * nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;-1' I I ¦ ¦nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;H ¦ I ¦

gt;4034«7P

. nbsp;nbsp;nbsp;nbsp;i4lt; 1 fprefid. (Cub.

Ct9O nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;_

14389000. max.

De Probationibus inuen*

*• ’ ¦ tioiu's Cubicx

radicis.

Hibet amp; ilia tres experientias. MuP tiplieationem nouenau’am fcptcnalt;nbsp;riam.

Déprima Duc radicem infcCubicc. €lt;rcfiduâ,fi adfit,addeprcducfto, 2C darnbsp;tus rcdibit numerus

Deiecunda. Sume probam de Quo' tiente.

Epitome 5. LX XV. tJetc, qviâminfccubiccîx^pî'Qdudtopronbsp;bamrofiduiaddc, ódabiccflis nouem iclinbsp;ôumadanjllcrucispûnej c uidatxnu-pieriproba par erft,

* D c tertia. Cum hac vt nouenaria agi-tofuis tarnen legibus fcruatis.

Exemplum.

* nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;*nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;r

. loiSrcfid. 57966(3?

Arithmetic«

De Fradionibus feu parti' busintcgronim.nbsp;ICap.L

Ntegronim ratio haclenus vifa,quoriî partes Minutiæfeufradiones dicunturnbsp;Et plane nihil aliudfunt fradiones, quînbsp;Diuifi onis refi duum

¦ FraClio eft aliqua pars integri .Pars autem aliqua dicitur quæ aliquoties tc'nbsp;petitatotum conftituit.

Vnitatesnumcrorû hic pro partibus integri fumuntur.

Idem ad diuerfacollatum dicipotcft, integrum iuxta ac fracfl'iOj vt minuta rc'nbsp;fpeiSu horæ: fecundi,nbsp;vulgaris feu Mer- Simplexnbsp;P A- catoria, cuius Mixta (dionbsp;fpcciesfunt, Fradionisfranbsp;onua-lia

Aftr0nomica,dequafuoIoco.

Simplex dicitur cuivnica in redo eft denominatio.vtfduæ tertiat.

•Mixta quæ diuerfos in redt o denomi natores Habet, vt hoc çft duæ tertiär,nbsp;tres quartæ quatuor quintac.

Frat^ïionis fraô-io duas ad tninirnuDe nominationes habet,quanim priorin Conbsp;loredo.Cæteræ fi plures funt,oinnes innbsp;cbüquoponuntur. vt, “hoceft vnaternbsp;tia vnius quartar vnius nicdietatis.

. FraÔionum integrorum eædcin

Psÿ[vmcratio hoc ïoeoeft débita Fra-ôionum repræfentatio, inhac duo funt numeri Superior qui num erator. Inferior qui Denominator vocat. Inter vtrûqjnbsp;linea médiat vt| Numerator eft qui ng-rnerum partium id eft quot fint partes o-ftendit.

Denominator eft numerus qui in quor partes integrum fit diflcdtum indicat.

Än'tbmcticcs

Vt duas fcrtias ita numcrarc pfotensf fuut autem dux tertiæjdux partes uniusnbsp;inregrt in tres diuifi,

Fralt;flioms fra(fî:io I'ta rcprxfcntatur, vtfra(?lio,quæ m redo eft, finiftram ucrnbsp;fus ponatur, inter cuius numeratorcm Sdnbsp;denoniinatorem li'nca médiat. Fradio-ncs autem aliac, quarum denominatoresnbsp;in obliquo funt, dextram uerfus abftp ünbsp;ncamediantcloccntur,ut J ƒ J id eft, tresnbsp;quintacjuniusfccundæ du ar um tertiarü.

Inuemtur ah'quoties mixta fradicn» f radio, hxccftqux plures fradionumnbsp;fradiones intercipït,vt dux tei tiæ vniusnbsp;mcdictaus, quatuor quintx duarum ta?*nbsp;tiarum.

* t »

3 * T gt; nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;..

Canones numeratiom’s. »

Si numerator xquah's eft dénomma* tori. Minutia integrum praedfeconftF-tuit.vt, H

Si numerator Denominatore tnaipj eft, Minuda plus integro facit.

S i numerator Deno m fnatorc mi nor eft, Minutia minus integiorepraefentaC

Defraeftionum Rcdult;ftionc Cap.III.

Jr nisfint,adfeaddinonpoflunr.

Fia(ft:ones diuerfarum Denomina* tionum funr quT diuci fos habent Denonbsp;minatorcs.Eiufdcm vero Denominationbsp;nisquaccundcmhabcnt.vt,-

Canones Reduftionis.

Duas dilTimilmm Deneminatorum fradionis advnum itarcduciro,Duc de*nbsp;nominatorcs infe, S)C produdum communis erit denominator vtriufcp fa licetnbsp;fradionis. Poftea numeratorem vnius

L S par

Arfthmctices

per denominatorem alteriusmultiplica ô^produdum fuonumeratorifupra ponas tta \

Si ucro fradioncs pïures fuermt,duas priores primum, ut didum eft abfólur.s,nbsp;amp;' ex utrocj numeratore vnum conftitunbsp;asita 17

B

i.

3

lamcumprodudoamp;^ tertiàfraâ-fonc iuxtaprimain operare rcgulam. vtfintnbsp;rcducendæf Duabus prioribus abfo-luitis fciltcet ex duabus tertijs ÔC tribusnbsp;quartis ^Cum in hoc igitur produdo QCnbsp;tcrtiafraAionefecundum primam regtf

Epitome LXXVIIE ergo|||faciunt fl’- quae italocentur.

3 i i

.. Î 4 «

A ..

6q

Ita etfi fraóliones quatuor fint cufö' tri’um priorum produêto dCquarta mi*'nbsp;nutia iuxta prima operate reguhm, SCnbsp;fic in alqs agendum.

Fradiones fradionum ad fimplices' minutias itareducito. Multiplica Si nü'nbsp;mcratoresamp;^ denominatorcs in feita'fnbsp;fcciuntl^

Integra in fracfiiones ita fóluas, duc nii* merura integrorum in denominatoremnbsp;minutia’formandæ IIt'Jfaciuntp, '

Frad ioncs adintegrafic reducito,di* uide numeratorem per denominatoremnbsp;amp;in quotientc numcrum integrornha*nbsp;bcbis.

F racfiionem craflam infubtiliorcm' ita transfer. Numeratorem craflae ducnbsp;in denominatorem fubrrhori?, Si pro-dutflum

.Anthmcticca

duÄum diuidc pcrcraflæ denominator rem SC quoties quacfitum often det. Refinbsp;duum,fifuerit, denominabitur aDeno*nbsp;minatorcQiiotientis, vt j faciunt 40 fe-xagefimas,

Deadditionc^

Cap. III.

pRacftionum igitureiindé denomint torcm habentium Numeratorcs tantumnbsp;adfeaddâtur, ô(Sproduc?to fubfcnbaturnbsp;Denominator, vt. J^^fadunt

Sifracîioncspluresquàm duæ fuennf, (uxtafccundam redui^ionis regulam o-peraberis,et reduétione omniû fa/ta,nanbsp;meratores fimph'citcraddes.

Si fractions fracftijncs addendacfinl fîmpliei fl aeftioni. Bas iuxta tertiam rC'nbsp;gulamreducito. Deinde cum produdonbsp;redudionis QC fi.mplid fratftione agas amp;nbsp;cundum tcnorcçn.prirnæ régulai.

Ff aclioaes integris vel e contra fie ad das«

Epitome LXXIX 4as.Duc num Crû inrcgrorS in Denomi-tiatorcm fradionis,5C producfio Numt-iatorcm addas,5C operatioriis tux Numerators habcbis, cui denommatorenlnbsp;inuariatum fubtjcias.

Defubtradi'one.

Cap.V»

Egula generalis eft, æqualem ab æ-quail 66 minorem a maiore poffe fûbtra-hi.maiorem vero à minoreneuti^fralt;fli ombus autem cuius maior eft numerator (redueftlone fada^eadem quoep maior dicetur fradio. cuius numerator minor, fradio quoep minor.

Redudionefada, num eratorcm minore à maiore fubtrahas, 66 refiduû pone pro numcratore relido vt|a Jmanet?.

Minutias abintegris itafubtrahito.

Pone integrum vt fradioncm per vnita tern

An'tlrmcticcs

tem fuppoCtâ.Multiplica deinde luxta pnmain rcdudioius regiilam SCredudltnbsp;onefada,Subtraheminorem niimeratonbsp;rem à maiori vt | ab ^remanent^,

Fracflionum fradîioncs à fimplici fra lt;9:ioneita auferas. Age primo luxta ternbsp;tiam Redu(£rionisregulatn,hac redudinbsp;onefaifla, cum produdo $gt;C fimplici fra*nbsp;dioneagasiuxta prima rcdudionis rc'nbsp;gulam vt abj remanent «5 hoc eft vnanbsp;tcrtïa.

De Multiplicatione-CaputVL

Duc numeratores dcnommatorcs in fevtf’faciunt^j.

Fradiones cum integrishoc pado tnttltiplica. Soluatur integrorum numenbsp;rûs iritinitatem fubfcnbendam. Deindenbsp;atin fradiom'bus fimph'cibus multipb',nbsp;ca,utj Ifacxunt®

Epitome LXXIX DcDiuifionc.

Caput Vli.

Î n firadionttm Diuffionc, DiuiTor dex-tram uerfus, diuidcnda autem fra(flio fi-niftram uerfus ponatur, Deinde numéro tor diuidendi in denominatorem diuifonbsp;tis ducatur, produdhim ent operationnbsp;hl's numeràtor, poftea denominator di'nbsp;uidendipernumcratorem diuiforis mulnbsp;fiph'Cetur,amp;^prödudum erit, ut 3 perjfinbsp;unt| '

Notandum quod fradiones muïtipïi Cando decrefeunt, fedcrefeunt diuiden-do. Et hoc cotra vocum naturam efleui-detur,ut fimultiplico f proueni't^, quænbsp;^ïaêïiû multo minor eft f aut At ft diuinbsp;do^ per multo maior minutia quam fnbsp;auti

Fradiones in 3. uel j aut in fnbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;aliani

~ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;An'thmetiees

aîratn tta rcfolues. Numerator? fra Aio*' nis diuidcndæ diuïdc,fi potes, per frat^i.nbsp;onem in quam transferre uoluen’s, Suf fubnbsp;quoticnrcm pone denominator? fradi-onïs diuidendæ ut * in x .faciunt . Itemnbsp;fin; faciunt gt;.Siucró id non polfi'Sjducnbsp;ergo denominator? fradfi'om's diuiden'nbsp;di”n numcratorem fradfioni's in quamnbsp;transferre diuidendam uolucris, 5^prO'nbsp;dudfum crit,d.nominator, numeratorcnbsp;inuariatour|in ƒ faciunt,

De Radicum Inuentione, Cap.VIII,nbsp;pRadfionum propofitara, antequaninbsp;radix quadrat a quæratur, eandcm cffenbsp;denominationem oportet,Quo cxifteiinbsp;tc,r3 dix,ut in integiis, quaeratur R àdixnbsp;numeratorum inuenta erit numerans denbsp;nominatoiisautem, denominans, ut 15«^

Epitome nbsp;nbsp;nbsp;LX XXL

* Sc I in additione coftituunt ^Radix numeratoris quadrata ( x i refid. x 8. Radix denominaforis( i ; refid.i j.

Fradiones autem, qua rum cubicam radiccm quæris , ad eandem denomina-tionem rcducito. Quofacflojducdeno-minatorem in fe quadrate, QC produAunbsp;turfumpcrcomunem numeratoremuLnbsp;tiplica, çin'ustande produdi cubica radinbsp;cem, vt in integris, quaere, quæ muêta 13.nbsp;dixerit numeratoris. Simpliciterautê,nbsp;radix cubic denominatoris, ueftigaturnbsp;vt ^’*1 Hæ reduc'læ Süfadditæ ad fe con-ftituunt rjr.Denominator in fe quadratenbsp;dudus producit 18 8743 (gt; 8. Huius pro-dudiradix cubica eft ( 5,refid. 67.

DeFradionibus Aftronomicis

1. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;M Ad

4

Anthmeticcs

Adcœleftïum orbiumcurfusexadi fupputandos inuenta funt quædâ inte^nbsp;gra ÔC eomm fradioncs. Hæc tarnen nrnbsp;tegra, maiorurefpedu,partes feufracflinbsp;ones did poflunt. Difponuntur autemnbsp;ita vt primus locus fit totius reuoliitioisnbsp;quæ J1 fl gna continet. Secundus figno'nbsp;tum.Tertiusgraduum. Quartusminu^nbsp;torum,fecundorum,quintusj fextus terlt;nbsp;tioru amp; fic ad fcptima vfcp progrefiio ßtnbsp;Signantur numeri vt in tabuli's Alfoi*nbsp;fiScaliorum,priori fradionis literaudnbsp;denominatore, vt T,s,g,m,sjt ,qr.

Denominator minutonim eft vnitas Secundorumbinarius, T ertiorum ternanbsp;riusjSCc.

Hue ctiam pertinent temporû fe(ftilt;t ncSjVt annus diuidit in i i. menfes, Mennbsp;fis in dies x s. 3 o vel 31. Dies in borasnbsp;Hora dertic^ in fuas per fexagenariamnbsp;diuiftonem partes fecatur.

DC

Epitöme LX XXII, Dereduólionc.

Fraclioncs omncs tam fabtrahendi «ïuàmeiuâquofitfubt radio,tam mul-tiph'candi multiphcantis, tam præte-réa diuidendi' diuiforis,prius ad eandênbsp;( fi non fint ) denominatorem reducen-dæquam ad operationcm conferantur*nbsp;IcgcSiScanonesinlibelIo,nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;-

De Additione,

Cap «IL

J—Jicc vt in integrîs fitjnifi bac cautjone feruata vt fradiones eiufdem dénommanbsp;tionis adfeaddantur,minuta fcilicet mLnbsp;iiutis,fecundafecundis ôdc.

- In Additione incipicndirm eft a fubti h’oribus, vtputaquartiSjfivltimaincx-emplo fint , procedcndumc^ finiftiamnbsp;yerfus ad tertia, deinde a tertqs ad fecunlt;nbsp;daSCcaêtcra. Etquotiesex additiqne 6 qnbsp;Mi

Arithm'etîccs prouenerint pro l'Ilis vnum fcqucnti fininbsp;¦ftram verfus craffi ori addatur.Et obfer'nbsp;uatur id vfcp ad gradus ex clufiue. Si ad-dito in gradibus cft loco j o graduum vrnbsp;nitasfequenticraffîorifracfiioni ( fignisnbsp;fcilicetjadijcit.Porrofi additio inügnisnbsp;cft, I zfignonimlocoponaturvnitasr»nbsp;tam reoolutioncm.

Exemplum,

SCap.III.

Vbtracîîioquocp vtin integrisfit inb tium præterea, ut iam in additionedilt;fi^nbsp;eft,

Epitome LXXXIIL eft, âfubtilioribus fumi'tur, £Cminutaânbsp;minutis,fecundaâ fecundis auferuntur,nbsp;Sgt;Cc.

Quod fi in fubtib’orum Tubtracftionc numenisa quo dcbetfieri fubtralt;ftio,fubnbsp;trahendo minor fucn’t, vnitas a proximanbsp;crafTiorcfiniftrâ verfus accjpiatur,qaacnbsp;in 6 o. pornones fracftioms minons diuPnbsp;dcndaeft,vtfubna(ftio fieri poffit.

Si in gradibus operaii ncqucas,vnum fignum in 30 giad.refoluendua fignisnbsp;accipias.

Si op er an'o in fignis impediatur, vna totareuoIutioC » 1 fcil.figna)mutuctur.

Tn tempowm fratftionibus fuæ quotp conditiones obfcnicntur , quæ inhuncnbsp;modum proponi pofl'unt.

Seculum,Indidio,Luftrum, OlyiH' pias, Ännus,Menfis,Dies,Hora, Minunbsp;turn, Secundum ,Tcrtium, Quartum

M SeculS

Aritlimcticcs

centum arinorff,. quindecimannQ,nbsp;j.anno*

4. anno.

iz mêfiöuel ^6jf àierûdC 6 ho.rarSnbsp;*8.50,31 dieru»nbsp;*4horarum,nbsp;óominutorumnbsp;é o fecundoru, Sinbsp;fic dcalqsperóo,nbsp;fubtradlionis,

G mi za zó 31 4Snbsp;i4

I »I i nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;i 4nbsp;nbsp;nbsp;53

De Mul tiplicatïone. cap.nn.

Hacc ita fit, Numerator in numerator^ ducit, Sc producïu dicit. Fradio â nume^nbsp;ro coiuncftorS denominators dcnomïnïnbsp;da,ut minuta in minuta duda producötnbsp;fecun

Epitome LXXXIIII (ccunda .Minuta mültiplfcata per tertianbsp;producunt quarta,8(^c. Quisautem denominator dici debeat, didum eft capirnbsp;teprimo.

Si fradiones in integra ducant no integra côftitauntur,fed fradiones,hoc eft, ftibtilior fradio ex integrorS multiplicanbsp;tione producit, vt minuta per gradns ftnbsp;multiplices, non gradus fed minuta effi-cies.minutaperfecudamultiplicatapronbsp;ducant fecunda amp; fempcr craffa fubtilionbsp;tem conftituit,

S gr» nbsp;nbsp;nbsp;nbsp;gr rni.

ƒ64; nbsp;nbsp;nbsp;loo 30

‘ gr. mi. ifiOnbsp;nbsp;nbsp;nbsp;nbsp;^ooo

DeDiuifîonc. Cap. V.

Tn Diuifione numerus quoties fra-dionis denominandus eft à numero qui proucnit poft fùbtnidioneia

Aï 4

Arithmeticcs denomùiatoris diuidendisâ dcriomina^nbsp;torediuidendijVt 1140 quartaper i o.fe-rnbsp;çundasin quötientc 4 fecundahabcbis..nbsp;Hoc eft, qaoties nominator à reh'do di-uifonsâ diuïdendi denominatore.

Siæqualia denominationcpcr æquaîi a di'm'daSjin quotiête non fradiones fednbsp;integra habcbis, vthorarum minuta pernbsp;minuta multiplïcata producunt horas.nbsp;Secundai'nfeeunda dudlafaciunt min.,nbsp;Hoc loco folaquotientis intrïnfeca denbsp;nominatio confideranda eft, l'dcft ,annbsp;figni'ficet fignaigradus,min .uel fecundanbsp;QCc. Vndefciendum quod l'ntn'nfeca de^nbsp;nomi'natiofumiturà denominatore,ex-trinfeca vero à numeratorc.

De Radicum inuentionc.

Cap. VI.

F

A Radtoncs, quarum petis quadratam fàdicem,prius,yt didhim eft, adeandem.nbsp;denomi'

Epitome LXXXV, dcnotninationem reduato,Qiiod fi eiufnbsp;dem dtnominationis, fed ab imp an nult;nbsp;mèro denominatæjfint ad candem dcno.nbsp;minationcm paris numeri reducaSjQuonbsp;faclo,ageficut in intcgris docuimus.Cænbsp;tcrum radix inuenta fignificat fradio'nbsp;nes a media fradiohe, integra vcrfus dclt;nbsp;nomina das. Media quidem dicitur, quxnbsp;inter radicis inuentæfradioncm amp;: inte-grum médiat vt fi a i ö 3 quartis radicenbsp;extrahas.f 16 pro radice et 7,pro refiduonbsp;habebis . At 1 6 a media fradione integra uerfus appellantur,fcilicet,a z.fecû-:,nbsp;dac. Nam fecundorû locus hoc locome-diuscft,vtgr. mi.ia. 5a.4a. Hoceft, radix inuenta fub duplam d en ommat io-nem eflentialem habebitr^rfpedu illiusnbsp;cuius radix quadrataquaphtur.

' nbsp;nbsp;nbsp;nbsp;nbsp;¦ M ÿ

Arithmeticcs

quot; nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Pàrjvtift

Fralt;îïioftum AftronomicarS quarta 6 a àlïæ dpnu'nantur à numero

Impari vt

mi. j.j.r Porto radix cubicajUt in integris quæ-ritur, Vcrum inuenta denominanda eftnbsp;atertiapartepropofitaefradHonis. Pro-inde fradiones,quarum cubica radix in-ueftigatùr, ad eandem denominationcnbsp;quac intreis partes aequaliter diuidipof-fitjrediganturvtradix zymi. eft 3 , no-ûorum.Namnouemfunttertiapars, ijr

' HadenusdeminutiiTimi« partibuSj quibusnbsp;Aftronomi

vtuntur*

Pc

Epitome LXXXVI. De Supputationc quae fit in Abaco,nbsp;Cap.I.

A Ba eus vulgo menfa dicitur calcula--^^toriz quibufdam diJ^ncfïalineis. In Abaco tria notâdailint. Primonbsp;^uplicesineo lineæfunt quarûaliæpa-fallclæjaliæ dicuntur orthogonales.

Parallelac funt quæ â dextralîniftram verfus protratllæà feæqualiterdiftant,nbsp;Harum officium eft reprefentare ziphranbsp;rum locahoc modo, infima hoc eftpri-«nalinea monadicum oftendit. Secunda

Orthogonales funt quæab imalinca ad fummâ rc Aa protendunt, vnde £lt; panbsp;rallelas ad âgulos recîos intcrfecât .Hasnbsp;ob varias mônetarum appellationes adnbsp;diftinguendos viculos ôt:euitandam conbsp;fufîoneminuenerût, Secundo notandSnbsp;qgt; in Abaco duplicia fpaciafôt qugda.n.

A n’t hm c tl CCS

parallelïs conftringuntur, Sdvocanttir, domus. Qtiædam vcro interfccann'bu«nbsp;diftinguirntur lineis 8é dicuntixr vicuït,nbsp;Tcmo denicp notandum quodquartanbsp;linea ex paralellis mi'Iinarium fignifl-cansftellula û«^nterfeA'onis pundofi'nbsp;gnari debet, vt,

NCap. II.

Vmcratio, quæ calculi'sfit, cftai-Aiscp numcri fecüdum lineas Sgt;C fpacia

conr

ïineanrm amp; fpacïorum. Delineanim va lorèpræccdcnticap.di'dumeft. Calculus autem in fpacio pofitus quinquicsnbsp;plus fignificat, quä fi idem i'n lineaindc-fccnfu proxima poneret.Idem practcreanbsp;calc ulus ïn fpacio pofitus dimïdiu oftennbsp;dû calculiinfuperioiilineapofiti.vt '

NumciOjVt didum eft.pcrïineas Si. fpacïo difpofito, maximus primo expr(nbsp;»natur, vt,

Subtraólionis . . r initiu el-

t . 1. nbsp;nbsp;. . fe debet, in

Multipncationis amp;Diuifionisnbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;fiimmisnbsp;duos proximos ponunt viculos.Deindcnbsp;omncs calculi vnius viculitrâsferunt innbsp;cadem fpacia Ô^lineas alterius viculi-Hac fols eautione feruata vt pro quinegnbsp;«akulisin linca pofitis locetttrvnus in

proximSfuperius (pacia, ptodüoburs vc ro calculisin fpaci'o lacentib.ponatur v-nus in lineam in afcenfu proximam. vt,nbsp;Nurti.addendus Num.fuperior Süma

lt; Infubtraifbonequoèp duo calcuTonim ordines funt Supeiior et fubtrahendüsjcnbsp;quib. tcrtius (relicfïus fcilicct) fubtradli-öne fada elicitur, Subtradio ininfimia

Aiithmètiœs

rt dixi,locis fnitium fumit, Subtrahetidus. Superior. ReliVhis.

16(0

j Si in linea fubtraheie non poffîsjrcfol ue calculum in fuperiorcfpacio pofitumnbsp;in quinc^ vnitàtes quas in tuam ponito Hnbsp;neam,ôi fubtrahe. Si nero in fpado nonnbsp;poflîs fubtrahere, refolue calculum in funbsp;perfore linea pofitum in duas vnitares,nbsp;quas in tuum pone fpacium, SCfubtra'nbsp;hc. vt»

Exemplum deLinea.

Sub*

An'thmetices ’

QtTcmadmodum eleuatio in Addi'tt* one ita refolutio i'n fubtraclioe frequês gnbsp;De Multiplkatione.

Multïplïcatlo eft ex vnius uumeri ducflu ahcuius tertij ïnuentio.

In Multiplicatione multiplicansno in Abacü ponitur,fed mente fol et tenen’

In Multiplicationemub'pbcans con fîderîdus eftjan fcilicet par vel imp ar fitnbsp;Multiplicatïo in fummis locis incipitnbsp;Ad multiplicandum opus eftdigito^nbsp;lineis admoueatur ex ordine.

. Omnis linea, cui digitos admouetur, tiumerum digitumrepræfentat.

Si fummus calculus in fpacio ponitur digitus appliccturad lineâfuperiorem»

His ita confidcratis, pone multiplie» dum.adfuas lineas 6Cfpacia,óddigituinnbsp;fummælineæadmoueto.

Si mul tiplicans eft parex quolibet cal culo multiplicandi inlineapofito, totu^nbsp;cregioneeiufdê lineae conftituatur nnilnbsp;tiplicans,

Epitome nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;XC.

. Ek quollt)« autcm calculo fpacjum occupante, medietas multiplicantis refpe-lt;tu lineæ fuperioris ponatur, vnde digitus lineae adhærens non deponatur, donee fub eiufdcm lineæ fpacio calculus, fi quis adfucrit per multiplicationem ab-foluatur.

Sivero multiplicans eft impar,digt-tum vt prius ad lineam pone, Slt;:ex fin-gulis cal cub's in linea iacenribus, torutn c rcgioncmultiplicantem ponito, dem-deex fingqlisin fpacio fitis,medietatenbsp;maxtrni paris, qui inimpan’multipbcâ-te eft, ponas e regione cum dimidio vni-us qj fub ciufdcm lineæ locetur fpacio.

Abfolutis ergo omnibus calculis in It ïieaô^ eiufdem lineæ inferiori fpacio ponbsp;fitis,applica digitum in defeenfu fequennbsp;tilineæ,amp; vt prius agas,ita Ôl cum omninbsp;bus inferioribus operare lineis SC fpactjs,

Exemplum.pans,

De Diiiifione.

Diutfio quoc^ in fummis locis incipir, Siâ diuifor menti tenetur/digitufcp opponbsp;nitur.Ponatur ergo digitus adlineam innbsp;qua diinforemhaberepoiïît,nbsp;nbsp;quo ties

totu*

Epitome XCI. totusaufertur diuifor totics 0^ vm'tas cnbsp;regione ciufdem lineae, qua digitus tan-gitjlocctur. Debet autemin vnalineanbsp;quotiespoteft aufcrri. Deinde cum diginbsp;to tarn diu defeende donee diuifoi em itcnbsp;rum habere p offis, idtp in finein vnbsp;fcraetur,vt.

Exemplumvbi diuifor eft. t 8, Diuidendus. Quotics. Refid.

Exemplum aliud in quo DimTor eft. 7*. 6C nullum refiduum.

N r

J

An’thmcticcs

D/uidendus Quoties.

- • I

9

Haórenus de calculis.

De Regula Aurea AfiueTn'.

Rithmetici regulam quandam pro pter infinitum vfum vocant Auream,ó^nbsp;corrupte regulam deTrtquafi de tribusnbsp;numeris, vt quartus eliciatur nccefiarijsnbsp;vfcft numerus emptionis uel cmptæ reinbsp;numerus præcrj numerus quacftionisnbsp;vt enim i o per x. numis quanti cmo 0 a »nbsp;poma.) Canones.

Numerus cmptionisfiniftram verfus Iocetnr,Numerus quæft'ionis vero dex'nbsp;teram vcrfus, inter vtruni^ numerus pracnbsp;ctj mediet.nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;Quesna^'

Epitome XCII.

Quemadmodum numerus primus SC tcrtius, ita fecundus Qi. quartus per regu-lam inuenicndus nomine di. rc correfpo^ -dcant.

Nomineƒlt;^ena^¦us dcnario,vlna vinat. amp;ievt \vlnapanno,denariusaureo»nbsp;Proportio primi SC fecundi ea eft,quatnbsp;tertij 6lt; quarti, quæitem primi ôC tcnijnbsp;eft,eadem quorp fccundi ÔC quarti.

Numerisiuxta primamregulam di-fpoGtis, ducaturfecundusin tertium S£ produôum diuidatur per primumjamp;innbsp;quotiente quartus proucniet quæfîtus ,nbsp;vt 6 ouaemo 4 numis quanti emo 846 ?nbsp;oua. Secundus in tertium duólus facitnbsp;3384 .Sumaperprimumdiuifafacitiiinbsp;quotient e( ƒ 64 num os.

Si diuifor diuidendo maioreft, fran-gatur diuidendus in partes minores,vt ß diuidendus fit aureus, difibluatur in de-narios,ctuciatós num os aut obolos»

N 4

Arithmeticcs.

Si nnmcrus fccundiis frat^i'ones annexas habet, frangrantur ciufdcm name nintegrainfradiones eiufdcmdenominbsp;nationi's.

Si primus óótcrtius fradi'oncs iKibci, vtiufcpIntegrainfuasfoluantmmutiaü,,nbsp;/ Argentincnfcs.nbsp;tfol iFnburgenCcs.nbsp;\ 18 o )iŒnios^ Conftantïcnfes,

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De Inuentione Cycli Solaris Indidiom's èc

numeri.

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Quod

Epitome - XCin.

Quod fi diuifioncfacra, nihil Kcman' ferit diuifor quxfitum oftcndetlt;nbsp;An Annus currens fitnbsp;bifiextilis.

Nnos Chn ft I di Hide per 4 Sf in I'cfi-. ¦*^duo dcnominationem anni curren-tis inuenies .Quod fi poft operationcmnbsp;nihil rcmïfcrir,diuifor quæfitû oftédet.nbsp;Denominatioannicurrentis eft anfeinbsp;licet annus fit bifèxtilis,aut primus, autnbsp;fecundus auttertius poft bifextum.

Subferiptisverflbusfegregantur i Chri ftiani â ludæis totidem.nbsp;Nondü poena mina ad te déclinât Aeneas.nbsp;Rexfrapci cum creme bona datfignaFcrena. ió’nbsp;Anglia dat lires tibi Ixtas temporefaftasnbsp;Numero mente concepto æqualemnbsp;addeproduefto numerum ,quem volçs,nbsp;adqce,atoto pofteaproducto dimidiSnbsp;rcmouc, a dimidio item relidto æqualcnbsp;paulopoft additumfeiunge, ÓCfempcrnbsp;rel inquetuT medie tas numeri vl timo ad*nbsp;ditus imparTàcf ft*,4refiduum ’quoep im-*nbsp;pareritcumfemifle. N s

Aritbmeticcs

Acnigma de tefleratuni fiimma inquirenda.

Prof] ce femel duas tefleras, amp;pfoda* Ao quod in fumma caram fuperficie eft;nbsp;adde altcrius teflerae fummam quac innbsp;imafuperficielatct, vnS colligepro-diicïhim. Alteram deinde tefferamrecpnbsp;pefimul ód protjce, Sii.fummam quæ innbsp;(umma fuperficie apparet, adijeepriorinbsp;produtflo, amp; rurfus vnS producfhim collt;nbsp;lige.Id ergo produdum,vt xnigmapronbsp;pofitum folues, fi teflerarum numero»nbsp;quem in fumma fuperficie vides, fep temnbsp;âdiungas.

De occttlte inquirenda fuma pecuniæ vel al terius reinbsp;ciafdem denominbsp;nationis.

, Quæn'tur, quo nam pado fuma qua-Îiiam incerta propofitaquantitatem ip-iuscfi:ra numerationem fcireliceat, ita *gas, diebabenti, uteamnameretper

Epitome Xcnn. tria,numcratione fa(îa,quid fupcrfit, I'nnbsp;tcnoga, fl fupercft vnitas, figncs tibi 7»nbsp;Si duo remanent, figncs tibi bis 70 bocnbsp;eft 140. Quofaclo die habcnti, vt fum-mam per quincp numcret, ÔChacnume-rationefaÂa,toties fignabis 1.1. quotie#nbsp;vnitatem in reft duo flip c reife intellcxe-risjtandem die habenti, vt candem fum-mam perrnumeret^ôf quotvnitates,nxtnbsp;merationc faifta, remanferint toties i fnbsp;fignato.Summa deindeomnium figna-torum collige, â collecte» auferquoticsnbsp;potes I o ƒ ^refiduum oftendetfuramS-priusignotam.

In menfa anulum inueftigarc quern vtraquismanu,quonbsp;digito,quouetencatarticulo.

lube Arithmeticc gnarum afeadefi rftp, qui anulum habeat, numerarenu-merumcpnotarcj Adderc $ produiftuntnbsp;multiplicare per j . Adderepofteanu-merum digiti:ita tarnen vt dextræ ma-nus minimus piimtB fit » fîniftræ

Arithmctices

poll ex fiat fextus amp;c. lube totum dc/ti-demultiplicarcper i o. Addereprodu-’ artkulS, ita vt fi primo digiti habeatnbsp;articuloaddat t.Sifecundo.z.fitertionbsp;Articulas autem vngui proximus primus eft. Quaere omnium lam di'clorumnbsp;fummâàquax ofubtrahe, amp;rcmane-bunt tres figuræ,quarum prima f ordincnbsp;retrogrado feruato )articulum, S ecundanbsp;digitum, Tertiauero repracfentat perfonbsp;namordinefedentenbsp;nbsp;annulû habentê.

Abaratio inquirendi' annub'.

Pn'neipio quacritur habentis ordo, nu merus duplatur. Addunturprodudo 7.nbsp;Multipbcaturtotum per f pofteama-nuum numerus additur ita, vt, ft de x tranbsp;foerit,adijcitur ». Siftniftra 1. totum de-indcmultiplicaturper t o.Produdo numerus digiti adiungitur, itavt vtriuftpnbsp;manus pollcx primus fit, totum poftexnbsp;multiplicatur per 1 o . Additur produ-do arttculus, inter articulos autê vnguinbsp;proximus

Epitome nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;XCV»

proximus primus eft . Ab bac tandem fummafubtrahuntur 5 ƒ o e amp;refiduumnbsp;quatuor habcbit figuras quarum primanbsp;( ordinc rctragrado no tat o ) articulumnbsp;oftendit, Secundadigitum. Tertiamar-num,ôd Qiiarta habentem annulum.

Paradigma de CHartis.

lube in vtrantp manu Chartas aliquE accipere, ita vt numerus in vna manunbsp;par,in altera fit impar. Die deindevtnbsp;niusmanus( quam tu vis) Chartas o cul tenbsp;duplet, ôi^ duplato addat Chartas alteriusnbsp;Deinde feifeitare an produdum par fitnbsp;vel impar. Si par numerus illiusmanusnbsp;(quâvoluifti ) impar,fiveroprodudiînbsp;impar tuent, manus tua par erit.

Paradigma de TabelIiO' nibus.

Tabcllio

Arithmeticcs

Tabellioin fingulosdies 6 mil as ab' foluit eundo 5gt;^lam terttus poft fuum a-bnumdiesagitur,amp;ficperegit 1E milasnbsp;Quarto autrm die alius poft cuni mitti*nbsp;tur qui expeditius proficifccns fingulisnbsp;diebuspertranfit 8 milas . Quæriturer-'nbsp;go quor dierum fpacio fcquens præcc'nbsp;dentem tabellioncm attingat.Subtrahenbsp;é ab S 8Cmanent duo. Diuide igiiur i 8nbsp;per a SCinquûticntequæfitum habcbis.

Aliud.

Duo nunctj funt quorum alter à Fri-burgo abit Romam fingulis diebus 6, milasperagens. Eadem hora e Roma alnbsp;ter Friburgum proficifeitur fingulis^nbsp;diebus abfoluit S milas, Diftat autemnbsp;Friburgum Brifgoicum a Roma i o o,nbsp;mills, Quoigitur die ambo nuncij cou'nbsp;ucniunt.Addeó 8 faciunt 14, Diuidenbsp;» • o per 14 QC habebis qua:fitum vt 7

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Hofpitibus dcccm, diftribuuntur ïed-i nouem fic vt ßngulis Icóus ccdat.

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• 9 » • • 9 9 9 9 ^Contcntio inter duos deocca

Iinpreflum Argentorati perBar ptholetneS Gniningenim»nbsp;Anno. M D XXXVX.

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